Chapter 7 Determinants: Exercises - Linear Algebra with Python

Compiled from the original manuscript of Chapter 7 exercises provided by Professor Ping-Zen Ong: 36 top-level exercises in all, in the same order as in the manuscript. Every exercise comes with a hint and a complete solution that can be expanded; computational exercises are followed by a NumPy verification, while pure proof exercises have no accompanying program.

This part uses 0-based indexing, and permutations are written as lists of images. Inconsistent notation and ambiguities in the statements of the original manuscript are noted in the corresponding exercises.

import numpy as np
np.set_printoptions(precision=4, suppress=True, linewidth=100)

def print_header(title):
    print("=" * 60)
    print(f"  {title}")
    print("=" * 60)

def print_step(step, desc):
    print(f"\n▶ Step {step}: {desc}")
    print("-" * 40)

def compare_print(label, actual, expected_desc):
    print(f"\n[{label}]")
    print(f"  Computed value: {actual}")
    print(f"  Expected value: {expected_desc}")

def check(label, actual, expected):
    compare_print(label, actual, expected)
    assert np.allclose(actual, expected), label
    print("  check passed")

def rref(A, tol=1e-10):
    """Gauss–Jordan elimination with partial pivoting; only for the small numerical examples of this chapter."""
    R = np.array(A, dtype=float, copy=True)
    row = 0
    for col in range(R.shape[1]):
        if row == R.shape[0]:
            break
        pivot = row + np.argmax(np.abs(R[row:, col]))
        if abs(R[pivot, col]) <= tol:
            continue
        R[[row, pivot]] = R[[pivot, row]]
        R[row] /= R[row, col]
        for i in range(R.shape[0]):
            if i != row:
                R[i] -= R[i, col] * R[row]
        row += 1
    R[np.abs(R) < tol] = 0
    return R

def lu_no_pivot(A):
    """Only for the square matrices in this book whose first n-1 pivots are nonzero."""
    U = np.array(A, dtype=float, copy=True)
    n = len(U)
    L = np.eye(n)
    for j in range(n - 1):
        assert abs(U[j, j]) > 1e-10, "zero pivot: this function does not exchange rows"
        for i in range(j + 1, n):
            L[i, j] = U[i, j] / U[j, j]
            U[i] -= L[i, j] * U[j]
    return L, U

from itertools import permutations
import matplotlib.pyplot as plt

# --- Color palette (Accent Mix) ---
C_BG = "#F8F8F8"
C_GRID = "#D6D6D6"
C_AXIS = "#000000"
C_V1 = "#57068C"
C_V2 = "#006385"
C_T1 = "#2AD2C9"
C_T2 = "#8900E1"
C_WARN = "#FF5D47"
C_AUX = "#AB82C5"
C_AREA = "rgba(87, 6, 140, 0.25)"  # for Plotly only.
# Fonts: the English edition needs no CJK font, so nothing is downloaded when _lang == 'en';
# the Chinese editions use this same block to fetch Noto Sans TC/SC where no CJK font is installed
import os, urllib.request
import matplotlib.font_manager as fm
_lang = 'en'
_cjk = ['Microsoft JhengHei', 'PingFang TC', 'Noto Sans CJK TC', 'Noto Sans TC']
_have = {f.name for f in fm.fontManager.ttflist}
if _lang != 'en' and not _have & set(_cjk):
    _font = os.path.join(os.path.expanduser('~'), '.cache', 'fonts', 'NotoSansTC.ttf')
    try:
        if not os.path.exists(_font):
            os.makedirs(os.path.dirname(_font), exist_ok=True)
            urllib.request.urlretrieve('https://github.com/google/fonts/raw/main/ofl/notosanstc/NotoSansTC%5Bwght%5D.ttf', _font + '.part')
            os.replace(_font + '.part', _font)
        fm.fontManager.addfont(_font)
        _have.add('Noto Sans TC')
    except OSError as err:
        print('Could not download the CJK font; Chinese text in figures may not display:', err)
plt.rcParams['font.family'] = [f for f in _cjk if f in _have] + ['DejaVu Sans']
plt.rcParams['axes.unicode_minus'] = False

def inversion_count(sigma):
    return sum(sigma[i] > sigma[j] for i in range(len(sigma)) for j in range(i+1,len(sigma)))

def permutation_sign(sigma):
    return (-1)**int(inversion_count(sigma))

def cofactor(A, i, j):
    minor = np.delete(np.delete(A, i, axis=0), j, axis=1)
    return (-1)**(i+j) * np.linalg.det(minor)

Permutations, Inversions, and the Classical Definition¶

Exercise 1 (Inversions and Crossings)

On $\{0,1,2,3,4\}$ , let the permutation have the list of images $\sigma=[3,0,4,1,2]$ , that is, $\sigma(0)=3,\sigma(1)=0,\sigma(2)=4,\sigma(3)=1,\sigma(4)=2$ .

Compute the number of inversions and its parity.
Compute $\operatorname{sgn}(\sigma)$ .
Write $0,1,2,3,4$ in the top row and $3,0,4,1,2$ in the bottom row, and connect equal numbers in the two rows; count the pairs of crossing segments and compare with the number of inversions (the figure is in the output of the program for this exercise).
Associating 1 with even and -1 with odd, compare the three parities above.

Notation. Throughout this part, square brackets always denote a list of images, not cycle notation; the 1–5 of the original manuscript have been changed to 0–4 accordingly.

Hint. List all pairs with $i<j$ and $\sigma(i)>\sigma(j)$ .

# ============================================================
print_header("Exercise 7-1 | Inversions and Crossings")
# ============================================================

# --- Step 1 ---
print_step(1, "Compute the number of inversions and the inverse permutation")
sigma=np.array([3,0,4,1,2]); inverse=np.argsort(sigma)
check("number of inversions", inversion_count(sigma), 5)
check("sgn(σ)", permutation_sign(sigma), -1)
check("pairs of crossing segments", inversion_count(inverse), 5)
# --- Step 2 ---
print_step(2, "Draw the permutation diagram joining equal values in the two rows")
fig,ax=plt.subplots(figsize=(6,6),facecolor=C_BG)
ax.set_facecolor(C_BG)
ax.grid(True,color=C_GRID,linewidth=.8,alpha=.7)
colors=[C_V1,C_V2,C_T1,C_WARN,C_T2]
for value, bottom in enumerate(inverse):
    ax.plot([value,bottom],[2,0],color=colors[value],linewidth=2.4)
for position in range(5):
    ax.scatter([position,position],[2,0],s=75,color=C_BG,edgecolor=C_AXIS,zorder=3)
    ax.text(position,2.18,str(position),ha='center',color=C_AXIS,fontsize=13)
    ax.text(position,-.28,str(sigma[position]),ha='center',color=C_AXIS,fontsize=13)
ax.set_xlim(-.65,4.65); ax.set_ylim(-1.65,3.65)
ax.set_aspect('equal',adjustable='box'); ax.set_box_aspect(1)
ax.set_xticks([]); ax.set_yticks([])
ax.set_title('Line diagram of the permutation: 5 pairs of crossing segments, sign −1',color=C_AXIS)
for spine in ax.spines.values():spine.set_visible(False)
plt.tight_layout(); plt.show()

============================================================
  Exercise 7-1 | Inversions and Crossings
============================================================

▶ Step 1: Compute the number of inversions and the inverse permutation
----------------------------------------

[number of inversions]
  Computed value: 5
  Expected value: 5
  check passed

[sgn(σ)]
  Computed value: -1
  Expected value: -1
  check passed

[pairs of crossing segments]
  Computed value: 5
  Expected value: 5
  check passed

▶ Step 2: Draw the permutation diagram joining equal values in the two rows
----------------------------------------

Exercise 2 (Computing Signs)

Find the sign of each of the following permutations on $\{0,1,2,3,4\}$ :

$[4,1,2,3,0]$ .
$[0,1,2,3,4]$ .
$[1,0,2,4,3]$ .
$[1,2,3,4,0]$ .

Hint. For each position, count the images to its right that are smaller than its own image.

# ============================================================
print_header("Exercise 7-2 | Computing Signs")
# ============================================================

# --- Step 1 ---
print_step(1, "Compare the numbers of inversions and the signs one by one")
for sigma,count in [([4,1,2,3,0],7),([0,1,2,3,4],0),([1,0,2,4,3],2),([1,2,3,4,0],4)]:
    check("number of inversions", inversion_count(sigma), count)
    check("sign", permutation_sign(sigma), (-1)**count)

============================================================
  Exercise 7-2 | Computing Signs
============================================================

▶ Step 1: Compare the numbers of inversions and the signs one by one
----------------------------------------

[number of inversions]
  Computed value: 7
  Expected value: 7
  check passed

[sign]
  Computed value: -1
  Expected value: -1
  check passed

[number of inversions]
  Computed value: 0
  Expected value: 0
  check passed

[sign]
  Computed value: 1
  Expected value: 1
  check passed

[number of inversions]
  Computed value: 2
  Expected value: 2
  check passed

[sign]
  Computed value: 1
  Expected value: 1
  check passed

[number of inversions]
  Computed value: 4
  Expected value: 4
  check passed

[sign]
  Computed value: 1
  Expected value: 1
  check passed

Exercise 3 (Signs of Compositions)

Use the following three examples to test $\operatorname{sgn}(\sigma\circ\tau)=\operatorname{sgn}(\sigma)\operatorname{sgn}(\tau)$ (in the composition, $\tau$ acts first):

$\sigma=[1,4,3,2,0]$ , $\tau=[0,1,2,3,4]$ .
$\sigma=[2,3,4,0,1]$ , $\tau=[1,0,3,4,2]$ .
$\sigma=[1,4,3,2,0]$ , $\tau=[1,0,3,4,2]$ .

Hint. The $i$ -th image of the composition is $\sigma(\tau(i))$ .

# ============================================================
print_header("Exercise 7-3 | Signs of Compositions")
# ============================================================

# --- Step 1 ---
print_step(1, "Build the lists of images in the order of composition")
cases=[([1,4,3,2,0],[0,1,2,3,4],[1,4,3,2,0]),([2,3,4,0,1],[1,0,3,4,2],[3,2,0,1,4]),([1,4,3,2,0],[1,0,3,4,2],[4,1,2,0,3])]
for sigma,tau,expected in cases:
    comp=np.array(sigma)[tau]
    check("σ∘τ", comp, expected)
    check("sign of the composition", permutation_sign(comp), permutation_sign(sigma)*permutation_sign(tau))

============================================================
  Exercise 7-3 | Signs of Compositions
============================================================

▶ Step 1: Build the lists of images in the order of composition
----------------------------------------

[σ∘τ]
  Computed value: [1 4 3 2 0]
  Expected value: [1, 4, 3, 2, 0]
  check passed

[sign of the composition]
  Computed value: -1
  Expected value: -1
  check passed

[σ∘τ]
  Computed value: [3 2 0 1 4]
  Expected value: [3, 2, 0, 1, 4]
  check passed

[sign of the composition]
  Computed value: -1
  Expected value: -1
  check passed

[σ∘τ]
  Computed value: [4 1 2 0 3]
  Expected value: [4, 1, 2, 0, 3]
  check passed

[sign of the composition]
  Computed value: 1
  Expected value: 1
  check passed

Exercise 4 (The Classical Expansion)

The classical definition is $\det\mathbf{A}=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=0}^{n-1}a_{i,\sigma(i)}$ .

Verify that for $n=2$ it is $a_{00}a_{11}-a_{01}a_{10}$ .
List the six permutations in $S_3$ together with their signs, and use them to derive the formula for order three.

Hint. Each product takes exactly one entry from every row and every column. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 5 (A Sparse Expansion)

Use the classical definition to compute the determinant of

\mathbf{A}=\begin{pmatrix}0&2&0&5\\1&0&0&3\\2&0&4&0\\0&5&0&6\end{pmatrix}

(2)

Hint. First note that the only nonzero entry of column 2 must be chosen.

# ============================================================
print_header("Exercise 7-5 | A Sparse Expansion")
# ============================================================

# --- Step 1 ---
print_step(1, "List the nonzero terms of the classical definition")
A=np.array([[0,2,0,5],[1,0,0,3],[2,0,4,0],[0,5,0,6]])
terms=[]
for sigma in permutations(range(4)):
    term=permutation_sign(sigma)*np.prod(A[np.arange(4),sigma])
    if term:
        print(f"σ={sigma}: {term}"); terms.append(term)
check("nonzero terms", sorted(terms), [-48,100])
check("determinant", np.linalg.det(A), 52)

============================================================
  Exercise 7-5 | A Sparse Expansion
============================================================

▶ Step 1: List the nonzero terms of the classical definition
----------------------------------------
σ=(1, 0, 2, 3): -48
σ=(3, 0, 2, 1): 100

[nonzero terms]
  Computed value: [np.int64(-48), np.int64(100)]
  Expected value: [-48, 100]
  check passed

[determinant]
  Computed value: 51.999999999999986
  Expected value: 52
  check passed

Exercise 6 (The Reversal Matrix)

Find the determinants of the $n\times n$ identity matrix $\mathbf{I}_n$ and of the reversal matrix $\mathbf{P}$ ( $p_{i,n-1-i}=1$ , all other entries 0).

Hint. The reversal permutation makes every pair of indices an inversion.

# ============================================================
print_header("Exercise 7-6 | The Reversal Matrix")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the identity and reversal matrices of each order")
for n in range(1,9):
    check(f"det(I_{n})", np.linalg.det(np.eye(n)), 1)
    check(f"reversal matrix n={n}", np.linalg.det(np.eye(n)[::-1]), (-1)**(n*(n-1)//2))

============================================================
  Exercise 7-6 | The Reversal Matrix
============================================================

▶ Step 1: Check the identity and reversal matrices of each order
----------------------------------------

[det(I_1)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=1]
  Computed value: 1.0
  Expected value: 1
  check passed

[det(I_2)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=2]
  Computed value: -1.0
  Expected value: -1
  check passed

[det(I_3)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=3]
  Computed value: -1.0
  Expected value: -1
  check passed

[det(I_4)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=4]
  Computed value: 1.0
  Expected value: 1
  check passed

[det(I_5)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=5]
  Computed value: 1.0
  Expected value: 1
  check passed

[det(I_6)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=6]
  Computed value: -1.0
  Expected value: -1
  check passed

[det(I_7)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=7]
  Computed value: -1.0
  Expected value: -1
  check passed

[det(I_8)]
  Computed value: 1.0
  Expected value: 1
  check passed

[reversal matrix n=8]
  Computed value: 1.0
  Expected value: 1
  check passed

Exercise 7 (Triangular Determinants)

Prove that the determinant of an $n\times n$ upper triangular matrix is $\prod_{i=0}^{n-1}a_{ii}$ . Is the same true in the lower triangular case?

Hint. Any permutation other than the identity must pick up a zero outside the triangular region somewhere. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 8 (Determinants of Permutation Matrices)

Use the definition $(\mathbf{P}_\sigma)_{ij}=1$ if and only if $j=\sigma(i)$ .

Prove that $\det\mathbf{P}_\sigma=\operatorname{sgn}(\sigma)$ .
Using the composition property of the sign of a permutation, verify that $\det(\mathbf{P}_\sigma\mathbf{P}_\tau)=\det\mathbf{P}_\sigma\det\mathbf{P}_\tau$ .

Hint. Under this convention, $\mathbf{P}_\sigma\mathbf{P}_\tau=\mathbf{P}_{\tau\circ\sigma}$ . (This is a derivation/proof exercise; no Python verification is attached.)

Basic Properties and Computation by Elimination¶

Exercise 9 (Basic Properties)

Regard a determinant of order three as a function of its three row vectors, write out the classical six-term formula, and verify:

$\det\mathbf{I}_3=1$ .
With the other rows fixed, it is linear in any one row with respect to both addition and scalar multiplication.
Exchanging any two rows changes the sign of the determinant.

Hint. Each term contains exactly one entry from every row, and exchanging rows rearranges the permutation indices. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 10 (Computation by Elimination)

Use row operations to find the determinants of the following matrices:

\mathbf{A}=\begin{pmatrix}1&2&3&4\\2&3&4&1\\3&4&1&2\\4&1&2&3\end{pmatrix}, \mathbf{B}=\begin{pmatrix}3&1&1&1\\2&3&4&1\\1&1&2&1\\2&-1&-1&1\end{pmatrix}, \mathbf{C}=\begin{pmatrix}1&0&2&3\\0&1&1&2\\4&3&1&0\\2&1&0&1\end{pmatrix}.

(6)

Hint. If you only add multiples of one row to another, you can read off the determinant directly as the product of the final diagonal entries.

# ============================================================
print_header("Exercise 7-10 | Computation by Elimination")
# ============================================================

# --- Step 1 ---
print_step(1, "Cross-check the diagonal after elimination against the NumPy determinant")
cases=[([[1,2,3,4],[2,3,4,1],[3,4,1,2],[4,1,2,3]],160),([[3,1,1,1],[2,3,4,1],[1,1,2,1],[2,-1,-1,1]],0),([[1,0,2,3],[0,1,1,2],[4,3,1,0],[2,1,0,1]],-20)]
for A,expected in cases:
    L,U=lu_no_pivot(A)
    check("LU reconstruction", L@U, A)
    check("product of the diagonal of U", np.prod(np.diag(U)), expected)
    check("det(A)", np.linalg.det(A), expected)

============================================================
  Exercise 7-10 | Computation by Elimination
============================================================

▶ Step 1: Cross-check the diagonal after elimination against the NumPy determinant
----------------------------------------

[LU reconstruction]
  Computed value: [[1. 2. 3. 4.]
 [2. 3. 4. 1.]
 [3. 4. 1. 2.]
 [4. 1. 2. 3.]]
  Expected value: [[1, 2, 3, 4], [2, 3, 4, 1], [3, 4, 1, 2], [4, 1, 2, 3]]
  check passed

[product of the diagonal of U]
  Computed value: 160.0
  Expected value: 160
  check passed

[det(A)]
  Computed value: 160.00000000000009
  Expected value: 160
  check passed

[LU reconstruction]
  Computed value: [[ 3.  1.  1.  1.]
 [ 2.  3.  4.  1.]
 [ 1.  1.  2.  1.]
 [ 2. -1. -1.  1.]]
  Expected value: [[3, 1, 1, 1], [2, 3, 4, 1], [1, 1, 2, 1], [2, -1, -1, 1]]
  check passed

[product of the diagonal of U]
  Computed value: -1.1102230246251563e-15
  Expected value: 0
  check passed

[det(A)]
  Computed value: 0.0
  Expected value: 0
  check passed

[LU reconstruction]
  Computed value: [[1. 0. 2. 3.]
 [0. 1. 1. 2.]
 [4. 3. 1. 0.]
 [2. 1. 0. 1.]]
  Expected value: [[1, 0, 2, 3], [0, 1, 1, 2], [4, 3, 1, 0], [2, 1, 0, 1]]
  check passed

[product of the diagonal of U]
  Computed value: -20.0
  Expected value: -20
  check passed

[det(A)]
  Computed value: -20.000000000000007
  Expected value: -20
  check passed

Exercise 11 (Elementary Matrices)

For $n\times n$ elementary matrices ( $i\ne j$ ), prove:

The matrix $\mathbf{R}_{ij}(c)$ that adds $c$ times row $i$ to row $j$ has determinant 1.
The matrix $\mathbf{R}_i(c)$ that multiplies row $i$ by $c$ has determinant $c$ .
The matrix $\mathbf{P}_{ij}$ that exchanges rows $i,j$ has determinant -1.

For a scaling to count as an elementary row operation we need $c\ne0$ , but the determinant identity still holds when $c=0$ .

Hint. Apply the operations to the identity matrix, and use multilinearity and antisymmetry under exchange. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 12 (Row Operations and Multiplication)

For an $n\times n$ matrix $\mathbf{A}$ , prove:

$\det(\mathbf{R}_{ij}(c)\mathbf{A})=\det\mathbf{A}=\det\mathbf{R}_{ij}(c)\det\mathbf{A}$ .
$\det(\mathbf{R}_i(c)\mathbf{A})=c\det\mathbf{A}=\det\mathbf{R}_i(c)\det\mathbf{A}$ .
$\det(\mathbf{P}_{ij}\mathbf{A})=-\det\mathbf{A}=\det\mathbf{P}_{ij}\det\mathbf{A}$ .

Hint. There is no need to use the general product theorem first; simply apply multilinearity in the rows directly. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 13 (Effect of a Row Permutation)

For an $n\times n$ matrix $\mathbf{A}$ and $\sigma\in S_n$ , prove that

\det(\mathbf{P}_\sigma\mathbf{A})=\operatorname{sgn}(\sigma)\det\mathbf{A}=\det\mathbf{P}_\sigma\det\mathbf{A}.

(9)

Hint. Use adjacent exchanges to restore the rows to their original order. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 14 (Recording the Operations)

Let $\mathbf{A}=\begin{pmatrix}0&0&-1&4\\2&4&6&0\\1&2&1&1\\2&0&1&3\end{pmatrix}$ .

Find the determinant using row operations.
Record each elementary matrix, and write the ref in the form $\mathbf{E}_{k-1}\cdots\mathbf{E}_0\mathbf{A}=\mathbf{U}$ .
Find $\det\mathbf{U}$ .
Verify that $\det\mathbf{A}=(\prod_{i=0}^{k-1}\det\mathbf{E}_i^{-1})\det\mathbf{U}$ .

Hint. The work can be arranged with two row exchanges, whose minus signs cancel.

# ============================================================
print_header("Exercise 7-14 | Recording the Operations")
# ============================================================

# --- Step 1 ---
print_step(1, "Form the elementary matrices step by step and eliminate")
A=np.array([[0,0,-1,4],[2,4,6,0],[1,2,1,1],[2,0,1,3]],dtype=float)
E0=np.eye(4)[[2,1,0,3]]; E1=np.eye(4); E1[1,0]=-2
E2=np.eye(4); E2[3,0]=-2
E3=np.eye(4)[[0,3,2,1]]; E4=np.eye(4); E4[3,2]=4
U=A.copy(); det_factors=[]
for E in [E0,E1,E2,E3,E4]:
    U=E@U; det_factors.append(np.linalg.det(E))
check("U", U, [[1,2,1,1],[0,-4,-1,1],[0,0,-1,4],[0,0,0,14]])
# --- Step 2 ---
print_step(2, "Check the product of the determinants of the inverse operations")
check("det(U)", np.linalg.det(U), 56)
check("det(A)", np.linalg.det(U)/np.prod(det_factors), np.linalg.det(A))

============================================================
  Exercise 7-14 | Recording the Operations
============================================================

▶ Step 1: Form the elementary matrices step by step and eliminate
----------------------------------------

[U]
  Computed value: [[ 1.  2.  1.  1.]
 [ 0. -4. -1.  1.]
 [ 0.  0. -1.  4.]
 [ 0.  0.  0. 14.]]
  Expected value: [[1, 2, 1, 1], [0, -4, -1, 1], [0, 0, -1, 4], [0, 0, 0, 14]]
  check passed

▶ Step 2: Check the product of the determinants of the inverse operations
----------------------------------------

[det(U)]
  Computed value: 55.99999999999997
  Expected value: 56
  check passed

[det(A)]
  Computed value: 55.99999999999997
  Expected value: 56.00000000000002
  check passed

Exercise 15 (Factorizations and Determinants)

Prove:

Every unit lower triangular matrix $\mathbf{L}$ can be written as $\mathbf{L}=\mathbf{R}_0\mathbf{R}_1\cdots\mathbf{R}_{k-1}$ , where each $\mathbf{R}_i$ is a row-operation matrix of type $\mathbf{R}_{ij}(c)$ (adding a multiple of one row to another row).
If $\mathbf{A}=\mathbf{L}\mathbf{U}$ with $\mathbf{L}$ unit lower triangular, then $\det\mathbf{A}=\det\mathbf{L}\det\mathbf{U}=\det\mathbf{U}$ .
If $\mathbf{A}=\mathbf{P}_\sigma\mathbf{L}\mathbf{U}$ , then $\det\mathbf{A}=\det\mathbf{P}_\sigma\det\mathbf{L}\det\mathbf{U}=\operatorname{sgn}(\sigma)\det\mathbf{U}$ .
If $\mathbf{A}=\mathbf{L}\mathbf{D}\mathbf{U}$ with both triangular factors unit triangular, then $\det\mathbf{A}=\prod_{i=0}^{n-1}d_{ii}$ .

Hint. A unit triangular matrix can be reduced to the identity matrix by row-addition operations alone. (This is a derivation/proof exercise; no Python verification is attached.)

Block Determinants¶

Exercise 16 (Block Formulas)

Prove the following block determinant formulas, where the block sizes are such that all products are defined:

If $\mathbf{A},\mathbf{C}$ are square matrices of orders $p,q$ , respectively, then $\det\begin{pmatrix}\mathbf{A}&\mathbf{0}\\\mathbf{B}&\mathbf{C}\end{pmatrix}=\det\mathbf{A}\det\mathbf{C}$ .
$\det\begin{pmatrix}\mathbf{I}_p&\mathbf{B}\\\mathbf{C}&\mathbf{I}_q\end{pmatrix}=\det(\mathbf{I}_q-\mathbf{C}\mathbf{B})=\det(\mathbf{I}_p-\mathbf{B}\mathbf{C})$ .
If $\mathbf{D}$ is an invertible matrix of order $q$ , then $\det\begin{pmatrix}\mathbf{I}_p&\mathbf{B}\\\mathbf{C}&\mathbf{D}\end{pmatrix}=\det(\mathbf{D}-\mathbf{C}\mathbf{B})=\det\mathbf{D}\det(\mathbf{I}_p-\mathbf{B}\mathbf{D}^{-1}\mathbf{C})$ .
For blocks of order $n$ , $\det\begin{pmatrix}\mathbf{A}&\mathbf{I}_n\\\mathbf{C}&\mathbf{D}\end{pmatrix}=(-1)^n\det(\mathbf{C}-\mathbf{D}\mathbf{A})$ .

Hint. A block row-addition operation can be decomposed into ordinary row-addition operations, which do not change the determinant. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 17 (Block Computations)

Use the block formulas to find:

\mathbf{M}_0=\begin{pmatrix}2&3&0&0\\1&4&0&0\\5&-1&3&1\\2&6&1&2\end{pmatrix}, \mathbf{M}_1=\begin{pmatrix}3&1&1&0\\1&0&0&1\\1&1&2&1\\2&-1&-1&1\end{pmatrix}, \mathbf{M}_2=\begin{pmatrix}1&0&2&3\\0&1&1&2\\4&3&1&0\\2&1&0&1\end{pmatrix}.

(13)

Hint. Use, in order, the formulas for a block lower triangular matrix, for an identity block in the upper right, and for an identity block in the upper left.

# ============================================================
print_header("Exercise 7-17 | Block Computations")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the three block reductions")
M0=np.array([[2,3,0,0],[1,4,0,0],[5,-1,3,1],[2,6,1,2]])
M1=np.array([[3,1,1,0],[1,0,0,1],[1,1,2,1],[2,-1,-1,1]])
M2=np.array([[1,0,2,3],[0,1,1,2],[4,3,1,0],[2,1,0,1]])
check("block lower triangular", np.linalg.det(M0[:2,:2])*np.linalg.det(M0[2:,2:]), 25)
check("C-DA", M1[2:,:2]-M1[2:,2:]@M1[:2,:2], [[-6,-1],[4,0]])
check("D-CB", M2[2:,2:]-M2[2:,:2]@M2[:2,2:], [[-10,-18],[-5,-7]])
for M,expected in [(M0,25),(M1,4),(M2,-20)]:check("det(M)", np.linalg.det(M), expected)

============================================================
  Exercise 7-17 | Block Computations
============================================================

▶ Step 1: Check the three block reductions
----------------------------------------

[block lower triangular]
  Computed value: 25.00000000000001
  Expected value: 25
  check passed

[C-DA]
  Computed value: [[-6 -1]
 [ 4  0]]
  Expected value: [[-6, -1], [4, 0]]
  check passed

[D-CB]
  Computed value: [[-10 -18]
 [ -5  -7]]
  Expected value: [[-10, -18], [-5, -7]]
  check passed

[det(M)]
  Computed value: 25.000000000000007
  Expected value: 25
  check passed

[det(M)]
  Computed value: 4.000000000000001
  Expected value: 4
  check passed

[det(M)]
  Computed value: -20.000000000000007
  Expected value: -20
  check passed

Exercise 18 (Computing with the Schur Complement)

When $\mathbf{M}=\begin{pmatrix}\mathbf{A}&\mathbf{B}\\\mathbf{C}&\mathbf{D}\end{pmatrix}$ and $\mathbf{A}$ is invertible, $\det\mathbf{M}=\det\mathbf{A}\det(\mathbf{D}-\mathbf{C}\mathbf{A}^{-1}\mathbf{B})$ . Use this formula to find:

\mathbf{M}_0=\begin{pmatrix}1&2&3&4\\2&3&4&1\\3&4&1&2\\4&1&2&3\end{pmatrix}, \mathbf{M}_1=\begin{pmatrix}3&1&1&1\\2&3&4&1\\1&1&2&1\\2&-1&-1&1\end{pmatrix}, \mathbf{M}_2=\begin{pmatrix}0&5&0&2\\0&3&1&0\\2&0&4&0\\0&5&0&6\end{pmatrix}.

(14)

Hint. For $\mathbf{M}_2$ the upper-left $2\times2$ block cannot be inverted directly; exchange rows first.

# ============================================================
print_header("Exercise 7-18 | Computing with the Schur Complement")
# ============================================================

# --- Step 1 ---
print_step(1, "Compute the Schur complement with solve, avoiding an explicit inverse")
cases=[([[1,2,3,4],[2,3,4,1],[3,4,1,2],[4,1,2,3]],160,False),([[3,1,1,1],[2,3,4,1],[1,1,2,1],[2,-1,-1,1]],0,False),([[0,5,0,2],[0,3,1,0],[2,0,4,0],[0,5,0,6]],40,True)]
for values,expected,swap in cases:
    M=np.array(values,dtype=float); N=M[[0,2,1,3]] if swap else M
    A,B,C,D=N[:2,:2],N[:2,2:],N[2:,:2],N[2:,2:]
    S=D-C@np.linalg.solve(A,B)
    factor=-1 if swap else 1
    check("block formula", factor*np.linalg.det(A)*np.linalg.det(S), expected)
    check("direct determinant", np.linalg.det(M), expected)

============================================================
  Exercise 7-18 | Computing with the Schur Complement
============================================================

▶ Step 1: Compute the Schur complement with solve, avoiding an explicit inverse
----------------------------------------

[block formula]
  Computed value: 159.99999999999994
  Expected value: 160
  check passed

[direct determinant]
  Computed value: 160.00000000000009
  Expected value: 160
  check passed

[block formula]
  Computed value: -3.9650822308041406e-16
  Expected value: 0
  check passed

[direct determinant]
  Computed value: 0.0
  Expected value: 0
  check passed

[block formula]
  Computed value: 39.99999999999999
  Expected value: 40
  check passed

[direct determinant]
  Computed value: 39.99999999999998
  Expected value: 40
  check passed

Properties of Determinants and Expansions¶

Exercise 19 (Computing with Properties)

Let $\mathbf{A},\mathbf{B}$ be $n\times n$ square matrices with $\det\mathbf{A}=3$ and $\det\mathbf{B}=-2$ . Find:

$\det(2\mathbf{A})$ .
$\det((\mathbf{A}^2)^\top(\mathbf{B}^\top)^2)$ .
$\det(\mathbf{A}^{-1})$ .
$\det(\mathbf{A}^{-1}\mathbf{B}\mathbf{A})$ .

Hint. Multiplying the whole matrix by 2 scales each of its $n$ rows by a factor of 2.

# ============================================================
print_header("Exercise 7-19 | Computing with Properties")
# ============================================================

# --- Step 1 ---
print_step(1, "Test the formulas with concrete matrices of several sizes")
for n in [1,2,4]:
    A=np.eye(n); A[0,0]=3
    B=np.eye(n); B[0,0]=-2
    check("det(2A)", np.linalg.det(2*A), 3*2**n)
    check("det((A²)^T(B^T)²)", np.linalg.det((A@A).T@B.T@B.T), 36)
    check("det(A^{-1})", np.linalg.det(np.linalg.inv(A)), 1/3)
    check("det(A^{-1}BA)", np.linalg.det(np.linalg.solve(A,B@A)), -2)

============================================================
  Exercise 7-19 | Computing with Properties
============================================================

▶ Step 1: Test the formulas with concrete matrices of several sizes
----------------------------------------

[det(2A)]
  Computed value: 6.0
  Expected value: 6
  check passed

[det((A²)^T(B^T)²)]
  Computed value: 36.0
  Expected value: 36
  check passed

[det(A^{-1})]
  Computed value: 0.3333333333333333
  Expected value: 0.3333333333333333
  check passed

[det(A^{-1}BA)]
  Computed value: -2.0
  Expected value: -2
  check passed

[det(2A)]
  Computed value: 12.0
  Expected value: 12
  check passed

[det((A²)^T(B^T)²)]
  Computed value: 36.0
  Expected value: 36
  check passed

[det(A^{-1})]
  Computed value: 0.3333333333333333
  Expected value: 0.3333333333333333
  check passed

[det(A^{-1}BA)]
  Computed value: -2.0
  Expected value: -2
  check passed

[det(2A)]
  Computed value: 48.00000000000001
  Expected value: 48
  check passed

[det((A²)^T(B^T)²)]
  Computed value: 36.0
  Expected value: 36
  check passed

[det(A^{-1})]
  Computed value: 0.3333333333333333
  Expected value: 0.3333333333333333
  check passed

[det(A^{-1}BA)]
  Computed value: -2.0
  Expected value: -2
  check passed

Exercise 20 (Properties of Column Operations)

For an $n\times n$ matrix $\mathbf{M}$ , add $c$ times column $i$ to column $k$ ( $i\ne k$ ) to obtain $\mathbf{M}'$ . Use the fact that transposition does not change the determinant to prove that the two matrices have the same determinant.
Express this operation as multiplication on the right by a matrix, and prove the claim that way.
How do column scaling and the exchange of two columns change the determinant?
Do normalization, multilinearity, and antisymmetry of the determinant still hold when stated in terms of columns?
First perform one round of column operations and then one round of row operations to simplify $\mathbf{B}=\begin{pmatrix}1&1&1&1\\1&2&2&2\\1&2&3&3\\1&2&3&4\end{pmatrix}$ , and find its determinant.

Hint. Transposition interchanges column operations and row operations; for the last matrix, take differences of adjacent columns from right to left.

# ============================================================
print_header("Exercise 7-20 | Properties of Column Operations")
# ============================================================

# --- Step 1 ---
print_step(1, "Take differences of adjacent columns from right to left")
B=np.array([[1,1,1,1],[1,2,2,2],[1,2,3,3],[1,2,3,4]],dtype=float)
M=B.copy()
for j in range(3,0,-1):M[:,j]-=M[:,j-1]
check("after column operations", M, np.tril(np.ones((4,4))))
# --- Step 2 ---
print_step(2, "Take differences of adjacent rows from bottom to top")
for i in range(3,0,-1):M[i]-=M[i-1]
check("after row operations", M, np.eye(4))
check("det(B)", np.linalg.det(B), 1)

============================================================
  Exercise 7-20 | Properties of Column Operations
============================================================

▶ Step 1: Take differences of adjacent columns from right to left
----------------------------------------

[after column operations]
  Computed value: [[1. 0. 0. 0.]
 [1. 1. 0. 0.]
 [1. 1. 1. 0.]
 [1. 1. 1. 1.]]
  Expected value: [[1. 0. 0. 0.]
 [1. 1. 0. 0.]
 [1. 1. 1. 0.]
 [1. 1. 1. 1.]]
  check passed

▶ Step 2: Take differences of adjacent rows from bottom to top
----------------------------------------

[after row operations]
  Computed value: [[1. 0. 0. 0.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]
 [0. 0. 0. 1.]]
  Expected value: [[1. 0. 0. 0.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]
 [0. 0. 0. 1.]]
  check passed

[det(B)]
  Computed value: 1.0
  Expected value: 1
  check passed

Exercise 21 (Cofactor Expansion)

Choose any row or column and use Laplace (cofactor) expansion to find:

\mathbf{A}=\begin{pmatrix}0&0&0&4\\0&0&4&1\\0&4&1&2\\4&1&2&3\end{pmatrix}, \mathbf{B}=\begin{pmatrix}3&1&0&1\\0&0&4&0\\1&1&0&1\\2&-1&0&1\end{pmatrix}, \mathbf{C}=\begin{pmatrix}1&0&2&3\\3&0&6&0\\4&0&1&0\\2&-4&12&0\end{pmatrix}.

(16)

Hint. Choose the row or column with the fewest nonzero entries, and do not forget $(-1)^{i+j}$ .

# ============================================================
print_header("Exercise 7-21 | Cofactor Expansion")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the sparse cofactor expansions against direct computation")
cases=[([[0,0,0,4],[0,0,4,1],[0,4,1,2],[4,1,2,3]],0,256),([[3,1,0,1],[0,0,4,0],[1,1,0,1],[2,-1,0,1]],1,-16),([[1,0,2,3],[3,0,6,0],[4,0,1,0],[2,-4,12,0]],0,252)]
for values,row,expected in cases:
    A=np.array(values)
    result=sum(A[row,j]*cofactor(A,row,j) for j in range(4))
    check("expansion along a row", result, expected)
    check("direct determinant", np.linalg.det(A), expected)

============================================================
  Exercise 7-21 | Cofactor Expansion
============================================================

▶ Step 1: Check the sparse cofactor expansions against direct computation
----------------------------------------

[expansion along a row]
  Computed value: 255.99999999999991
  Expected value: 256
  check passed

[direct determinant]
  Computed value: 255.99999999999994
  Expected value: 256
  check passed

[expansion along a row]
  Computed value: -16.000000000000007
  Expected value: -16
  check passed

[direct determinant]
  Computed value: -16.000000000000007
  Expected value: -16
  check passed

[expansion along a row]
  Computed value: 251.99999999999994
  Expected value: 252
  check passed

[direct determinant]
  Computed value: 252.00000000000003
  Expected value: 252
  check passed

Recurrences and Polynomials¶

Exercise 22 (A Tridiagonal Recurrence)

Let $A_n$ be the determinant of the $n\times n$ tridiagonal matrix whose main diagonal entries are all 2, whose superdiagonal and subdiagonal entries are all 1, and whose other entries are zero.

Find $A_1,A_2,A_3$ and conjecture a general formula.
Expanding along row 0, prove that $A_n=2A_{n-1}-A_{n-2}$ ( $n\ge3$ ).
Find the general formula and verify the conjecture.

Hint. Column 0 of the second minor submatrix has only one nonzero entry.

# ============================================================
print_header("Exercise 7-22 | A Tridiagonal Recurrence")
# ============================================================

# --- Step 1 ---
print_step(1, "Compare the matrix determinants with the general formula")
for n in range(1,11):
    A=2*np.eye(n)+np.diag(np.ones(n-1),1)+np.diag(np.ones(n-1),-1)
    check(f"A_{n}", np.linalg.det(A), n+1)

============================================================
  Exercise 7-22 | A Tridiagonal Recurrence
============================================================

▶ Step 1: Compare the matrix determinants with the general formula
----------------------------------------

[A_1]
  Computed value: 2.0
  Expected value: 2
  check passed

[A_2]
  Computed value: 2.9999999999999996
  Expected value: 3
  check passed

[A_3]
  Computed value: 4.0
  Expected value: 4
  check passed

[A_4]
  Computed value: 4.999999999999999
  Expected value: 5
  check passed

[A_5]
  Computed value: 6.0
  Expected value: 6
  check passed

[A_6]
  Computed value: 6.999999999999998
  Expected value: 7
  check passed

[A_7]
  Computed value: 7.999999999999998
  Expected value: 8
  check passed

[A_8]
  Computed value: 8.999999999999998
  Expected value: 9
  check passed

[A_9]
  Computed value: 9.999999999999998
  Expected value: 10
  check passed

[A_10]
  Computed value: 10.999999999999996
  Expected value: 11
  check passed

Exercise 23 (A General Tridiagonal Formula)

Let $D_n$ be the determinant of the $n\times n$ tridiagonal matrix with main diagonal entries 3, superdiagonal entries 2, and subdiagonal entries 1.

Find $D_1,D_2$ .
Use Laplace expansion to prove that $D_n=3D_{n-1}-2D_{n-2}$ ( $n\ge3$ ).
Find a general formula for $D_n$ .

Hint. Let the successive difference be $K_n=D_n-D_{n-1}$ ; it forms a geometric sequence.

# ============================================================
print_header("Exercise 7-23 | A General Tridiagonal Formula")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the nonsymmetric tridiagonal matrices")
for n in range(1,11):
    A=3*np.eye(n)+2*np.diag(np.ones(n-1),1)+np.diag(np.ones(n-1),-1)
    check(f"D_{n}", np.linalg.det(A), 2**(n+1)-1)

============================================================
  Exercise 7-23 | A General Tridiagonal Formula
============================================================

▶ Step 1: Check the nonsymmetric tridiagonal matrices
----------------------------------------

[D_1]
  Computed value: 3.0000000000000004
  Expected value: 3
  check passed

[D_2]
  Computed value: 7.000000000000001
  Expected value: 7
  check passed

[D_3]
  Computed value: 15.0
  Expected value: 15
  check passed

[D_4]
  Computed value: 31.0
  Expected value: 31
  check passed

[D_5]
  Computed value: 62.99999999999999
  Expected value: 63
  check passed

[D_6]
  Computed value: 126.99999999999999
  Expected value: 127
  check passed

[D_7]
  Computed value: 254.99999999999991
  Expected value: 255
  check passed

[D_8]
  Computed value: 511.0
  Expected value: 511
  check passed

[D_9]
  Computed value: 1022.9999999999995
  Expected value: 1023
  check passed

[D_10]
  Computed value: 2046.9999999999995
  Expected value: 2047
  check passed

Exercise 24 (An Arrowhead Matrix)

Let $K_n$ be the determinant of the $n\times n$ matrix whose main diagonal entries are all 2, in which every entry of row $n-1$ and of column $n-1$ other than the diagonal entry is 1, and whose remaining entries are zero.

Find $K_1,K_2,K_3$ .
Find a general formula valid for all $n\ge1$ .

Hint. Eliminate the first $n-1$ entries of the last row.

# ============================================================
print_header("Exercise 7-24 | An Arrowhead Matrix")
# ============================================================

# --- Step 1 ---
print_step(1, "Check small sizes, the degenerate size, and negative values")
for n in range(1,10):
    A=2*np.eye(n)
    A[-1,:-1]=1; A[:-1,-1]=1
    check(f"K_{n}", np.linalg.det(A), 2.0**(n-2)*(5-n))

============================================================
  Exercise 7-24 | An Arrowhead Matrix
============================================================

▶ Step 1: Check small sizes, the degenerate size, and negative values
----------------------------------------

[K_1]
  Computed value: 2.0
  Expected value: 2.0
  check passed

[K_2]
  Computed value: 2.9999999999999996
  Expected value: 3.0
  check passed

[K_3]
  Computed value: 4.0
  Expected value: 4.0
  check passed

[K_4]
  Computed value: 3.999999999999999
  Expected value: 4.0
  check passed

[K_5]
  Computed value: 0.0
  Expected value: 0.0
  check passed

[K_6]
  Computed value: -15.999999999999998
  Expected value: -16.0
  check passed

[K_7]
  Computed value: -63.99999999999998
  Expected value: -64.0
  check passed

[K_8]
  Computed value: -192.0
  Expected value: -192.0
  check passed

[K_9]
  Computed value: -511.99999999999994
  Expected value: -512.0
  check passed

Exercise 25 (A Polynomial Matrix)

For $n\ge2$ , let the first $n-1$ columns of $\mathbf{H}(x)$ have $x$ on the main diagonal, -1 on the subdiagonal, and zeros elsewhere, and let the last column be $a_0,a_1,\ldots,a_{n-2},x+a_{n-1}$ from top to bottom. That is,

\mathbf{H}(x)=\begin{pmatrix}x&0&\cdots&0&a_0\\-1&x&\cdots&0&a_1\\0&-1&\ddots&0&a_2\\\vdots&\vdots&\ddots&x&a_{n-2}\\0&0&\cdots&-1&x+a_{n-1}\end{pmatrix}.

(21)

Prove that $\det\mathbf{H}(x)=x^n+\sum_{i=0}^{n-1}a_ix^i$ . When $n=1$ , this is to be read as the matrix $(x+a_0)$ .

Hint. Expand along the last column and find each cofactor. (This is a derivation/proof exercise; no Python verification is attached.)

Adjugates, Inverses, and LU¶

Exercise 26 (Adjugate and Inverse)

Let $\mathbf{A}=\begin{pmatrix}1&2&0\\0&1&3\\2&0&1\end{pmatrix}$ .

Find the nine cofactors $C_{ij}=(-1)^{i+j}\det\mathbf{A}_{ij}$ , where $\mathbf{A}_{ij}$ is obtained by deleting row $i$ and column $j$ .
Find the adjugate matrix and the inverse matrix.

Hint. The adjugate matrix is the transpose of the cofactor matrix.

# ============================================================
print_header("Exercise 7-26 | Adjugate and Inverse")
# ============================================================

# --- Step 1 ---
print_step(1, "Compute the cofactors one by one")
A=np.array([[1,2,0],[0,1,3],[2,0,1]])
C=np.array([[cofactor(A,i,j) for j in range(3)] for i in range(3)])
check("cofactor matrix", C, [[1,6,-2],[-2,1,4],[6,-3,1]])
# --- Step 2 ---
print_step(2, "Transpose, divide by the determinant, and check on both sides")
B=C.T/13
check("det(A)", np.linalg.det(A), 13)
check("A A^{-1}", A@B, np.eye(3)); check("A^{-1} A", B@A, np.eye(3))

============================================================
  Exercise 7-26 | Adjugate and Inverse
============================================================

▶ Step 1: Compute the cofactors one by one
----------------------------------------

[cofactor matrix]
  Computed value: [[ 1.  6. -2.]
 [-2.  1.  4.]
 [ 6. -3.  1.]]
  Expected value: [[1, 6, -2], [-2, 1, 4], [6, -3, 1]]
  check passed

▶ Step 2: Transpose, divide by the determinant, and check on both sides
----------------------------------------

[det(A)]
  Computed value: 13.0
  Expected value: 13
  check passed

[A A^{-1}]
  Computed value: [[ 1.  0. -0.]
 [ 0.  1. -0.]
 [ 0.  0.  1.]]
  Expected value: [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
  check passed

[A^{-1} A]
  Computed value: [[ 1.  0.  0.]
 [-0.  1. -0.]
 [ 0.  0.  1.]]
  Expected value: [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
  check passed

Exercise 27 (Cramer’s Rule)

Let $\mathbf{A}$ be an invertible $n\times n$ matrix and $\mathbf{A}\mathbf{x}=\mathbf{b}$ . Using $\mathbf{A}^{-1}=\operatorname{adj}(\mathbf{A})/\det\mathbf{A}$ and cofactor expansion, prove that

x_i=\frac{\det\mathbf{A}_i}{\det\mathbf{A}}, 0\le i<n,

(25)

where $\mathbf{A}_i$ is the matrix obtained by replacing column $i$ of $\mathbf{A}$ with $\mathbf{b}$ .

Erratum in the original manuscript. In the source, the math delimiters of the inverse-matrix formula are misplaced; the correct formula is $\mathbf{A}^{-1}=\operatorname{adj}(\mathbf{A})/\det(\mathbf{A})$ .

Hint. The cofactors of column $i$ do not depend on that column itself. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 28 (Leading Principal Submatrices)

Let $\mathbf{A}_k$ be the $k\times k$ leading principal submatrix in the upper-left corner of an $n\times n$ matrix (here $k$ denotes the size; entry indices still start from 0).

Use mathematical induction to prove: if $\det\mathbf{A}_k\ne0$ for every $1\le k\le n$ , then there is a unique LU.
Give an example showing that a unique LU does not guarantee that all leading principal minors are nonzero.

Hint. In the induction, separate the last row and the last column from the block whose LU is already known. (This is a derivation/proof exercise; no Python verification is attached.)

Area, Volume, and the Gram Matrix¶

Exercise 29 (Area and Order)

Let $\mathbf{A}=\begin{pmatrix}2&0\\0&1/2\end{pmatrix}$ and $\mathbf{B}=\frac1{\sqrt2}\begin{pmatrix}1&-1\\1&1\end{pmatrix}$ .

How does the area of the standard unit square $\Sigma=\mathbf{e}_0\wedge\mathbf{e}_1$ change under $\mathbf{A}\mathbf{B}$ and under $\mathbf{B}\mathbf{A}$ ?
What figures does $\Omega=\{(x,y):-1\le x,y\le1\}$ become under the two mappings?

Hint. Area depends only on the absolute value of the determinant, but shape is affected by the order in which the mappings act.

# ============================================================
print_header("Exercise 7-29 | Area and Order")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the composite matrices, the areas, and the side lengths")
A=np.diag([2,.5]); B=np.array([[1,-1],[1,1]])/np.sqrt(2)
for M in [A@B,B@A]:check("area scaling", abs(np.linalg.det(M)), 1)
check("the two columns of AB have equal length", np.linalg.norm((A@B)[:,0]), np.linalg.norm((A@B)[:,1]))
check("inner product of the columns of BA", (B@A)[:,0]@(B@A)[:,1], 0)
check("side lengths of the image rectangle under BA", 2*np.linalg.norm(B@A,axis=0), [4,1])
# --- Step 2 ---
print_step(2, "Check the vertices of the image under AB")
vertices=np.array([[-1,-1],[1,-1],[1,1],[-1,1]]).T
check("vertices under AB", A@B@vertices, [[0,2*np.sqrt(2),0,-2*np.sqrt(2)],[-1/np.sqrt(2),0,1/np.sqrt(2),0]])

============================================================
  Exercise 7-29 | Area and Order
============================================================

▶ Step 1: Check the composite matrices, the areas, and the side lengths
----------------------------------------

[area scaling]
  Computed value: 0.9999999999999998
  Expected value: 1
  check passed

[area scaling]
  Computed value: 0.9999999999999998
  Expected value: 1
  check passed

[the two columns of AB have equal length]
  Computed value: 1.457737973711325
  Expected value: 1.457737973711325
  check passed

[inner product of the columns of BA]
  Computed value: 0.0
  Expected value: 0
  check passed

[side lengths of the image rectangle under BA]
  Computed value: [4. 1.]
  Expected value: [4, 1]
  check passed

▶ Step 2: Check the vertices of the image under AB
----------------------------------------

[vertices under AB]
  Computed value: [[ 0.      2.8284  0.     -2.8284]
 [-0.7071  0.      0.7071  0.    ]]
  Expected value: [[0, np.float64(2.8284271247461903), 0, np.float64(-2.8284271247461903)], [np.float64(-0.7071067811865475), 0, np.float64(0.7071067811865475), 0]]
  check passed

Exercise 30 (Area in the Plane)

Let $\mathbf{u}=(2,2)^\top$ , $\mathbf{v}=(1,-3)^\top$ , and $\mathbf{M}=\begin{bmatrix}\mathbf{u} & \mathbf{v}\end{bmatrix}$ .

Find orthonormal vectors $\mathbf{f}_0,\mathbf{f}_1$ such that $\mathbf{f}_0$ points in the same direction as $\mathbf{u}$ (hence $\mathbf{f}_0^\top\mathbf{u}>0$ ) and $\mathbf{f}_1^\top\mathbf{v}>0$ .
Write $\mathbf{M}=\mathbf{F}\begin{pmatrix}h_0&r\\0&h_1\end{pmatrix}$ , and explain the geometric meaning of $h_0,h_1$ and why the area is their product.
Use orthogonality to prove that $\mathbf{F}^\top\mathbf{F}=\mathbf{I}_2$ and $|\det\mathbf{F}|=1$ .
Deduce that the area is $|\det\mathbf{M}|$ .
Derive $\det(\mathbf{M}^\top\mathbf{M})=(h_0h_1)^2$ directly from items 2 and 3 (without using item 4).
Explain why the procedure above generalizes to any two linearly independent vectors in the plane.

Hint. Project $\mathbf{v}$ onto the direction of $\mathbf{f}_0$ , then normalize the perpendicular component.

# ============================================================
print_header("Exercise 7-30 | Area in the Plane")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the orthogonal factorization and the sign conditions")
M=np.array([[2,1],[2,-3]])
F=np.array([[1,1],[1,-1]])/np.sqrt(2)
R=np.array([[2,-1],[0,2]])*np.sqrt(2)
check("F^T F", F.T@F, np.eye(2)); check("FR", F@R, M)
check("f0·u > 0", F[:,0]@M[:,0], 2*np.sqrt(2))
check("f1·v > 0", F[:,1]@M[:,1], 2*np.sqrt(2))
# --- Step 2 ---
print_step(2, "Compare base times height, the determinant, and the Gram determinant")
check("base times height", np.prod(np.diag(R)), 8)
check("|det(M)|", abs(np.linalg.det(M)), 8)
check("det(M^T M)", np.linalg.det(M.T@M), 64)

============================================================
  Exercise 7-30 | Area in the Plane
============================================================

▶ Step 1: Check the orthogonal factorization and the sign conditions
----------------------------------------

[F^T F]
  Computed value: [[ 1. -0.]
 [-0.  1.]]
  Expected value: [[1. 0.]
 [0. 1.]]
  check passed

[FR]
  Computed value: [[ 2.  1.]
 [ 2. -3.]]
  Expected value: [[ 2  1]
 [ 2 -3]]
  check passed

[f0·u > 0]
  Computed value: 2.82842712474619
  Expected value: 2.8284271247461903
  check passed

[f1·v > 0]
  Computed value: 2.82842712474619
  Expected value: 2.8284271247461903
  check passed

▶ Step 2: Compare base times height, the determinant, and the Gram determinant
----------------------------------------

[base times height]
  Computed value: 8.000000000000002
  Expected value: 8
  check passed

[|det(M)|]
  Computed value: 7.999999999999998
  Expected value: 8
  check passed

[det(M^T M)]
  Computed value: 63.99999999999998
  Expected value: 64
  check passed

Exercise 31 (Area in Space)

Let $\mathbf{u}=(1,2,2)^\top$ , $\mathbf{v}=(2,0,2)^\top$ , and $\mathbf{M}=\begin{bmatrix}\mathbf{u} & \mathbf{v}\end{bmatrix}\in\mathbb{R}^{3\times2}$ .

Find orthonormal vectors $\mathbf{f}_0\in\operatorname{span}(\mathbf{u})$ and $\mathbf{f}_1\in\operatorname{span}(\mathbf{u},\mathbf{v})$ with $\mathbf{f}_0^\top\mathbf{u}>0$ and $\mathbf{f}_1^\top\mathbf{v}>0$ .
Find $\mathbf{M}=\mathbf{F}\begin{pmatrix}h_0&r\\0&h_1\end{pmatrix}$ , and describe the base, the height, and the area.
With the $3\times2$ matrix $\mathbf{F}=\begin{pmatrix}\mathbf{f}_0&\mathbf{f}_1\end{pmatrix}$ , prove that $\mathbf{F}^\top\mathbf{F}=\mathbf{I}_2$ .
From items 2 and 3, deduce that the square of the area is $\det(\mathbf{M}^\top\mathbf{M})$ , and compute the area; explain why one cannot write $\det\mathbf{M}$ .
Check the area with the three-dimensional cross product.
Discuss the area formula, and the assumptions it requires, for any two vectors in three dimensions and for the generalization to $\mathbb{R}^n$ .

Hint. Project first; the general area formula holds, but the normalization requires a nonzero residual.

# ============================================================
print_header("Exercise 7-31 | Area in Space")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the rectangular orthogonal factorization")
M=np.array([[1,2],[2,0],[2,2]])
F=np.array([[1,2],[2,-2],[2,1]])/3
R=np.array([[3,2],[0,2]])
check("F^T F", F.T@F, np.eye(2)); check("FR", F@R, M)
check("f0·u > 0", F[:,0]@M[:,0], 3)
check("f1·v > 0", F[:,1]@M[:,1], 2)
# --- Step 2 ---
print_step(2, "Compare the Gram area with the cross-product area")
check("Gram", M.T@M, [[9,6],[6,8]])
check("Gram area", np.sqrt(np.linalg.det(M.T@M)), 6)
check("cross product", np.cross(M[:,0],M[:,1]), [4,2,-4])
check("length of the cross product", np.linalg.norm(np.cross(M[:,0],M[:,1])), 6)

============================================================
  Exercise 7-31 | Area in Space
============================================================

▶ Step 1: Check the rectangular orthogonal factorization
----------------------------------------

[F^T F]
  Computed value: [[1. 0.]
 [0. 1.]]
  Expected value: [[1. 0.]
 [0. 1.]]
  check passed

[FR]
  Computed value: [[1. 2.]
 [2. 0.]
 [2. 2.]]
  Expected value: [[1 2]
 [2 0]
 [2 2]]
  check passed

[f0·u > 0]
  Computed value: 3.0
  Expected value: 3
  check passed

[f1·v > 0]
  Computed value: 2.0
  Expected value: 2
  check passed

▶ Step 2: Compare the Gram area with the cross-product area
----------------------------------------

[Gram]
  Computed value: [[9 6]
 [6 8]]
  Expected value: [[9, 6], [6, 8]]
  check passed

[Gram area]
  Computed value: 6.0
  Expected value: 6
  check passed

[cross product]
  Computed value: [ 4  2 -4]
  Expected value: [4, 2, -4]
  check passed


[length of the cross product]
  Computed value: 6.0
  Expected value: 6
  check passed

Exercise 32 (Volume in Space)

Let $\mathbf{u}=(1,2,2)^\top$ , $\mathbf{v}=(2,0,2)^\top$ , $\mathbf{w}=(1,3,-2)^\top$ , and $\mathbf{M}=\begin{bmatrix}\mathbf{u} & \mathbf{v} & \mathbf{w}\end{bmatrix}$ .

Find an orthonormal basis $\mathbf{f}_0,\mathbf{f}_1,\mathbf{f}_2$ whose vectors lie, in order, in the spaces spanned successively by the first one, two, and three of the given vectors, and which satisfies $\mathbf{f}_0^\top\mathbf{u}>0$ , $\mathbf{f}_1^\top\mathbf{v}>0$ , and $\mathbf{f}_2^\top\mathbf{w}>0$ .
Write $\mathbf{M}=\mathbf{F}\mathbf{R}$ with $\mathbf{R}$ upper triangular with positive diagonal entries $h_0,h_1,h_2$ ; explain their geometric meaning and why the volume is their product.
With $\mathbf{F}=\begin{pmatrix}\mathbf{f}_0&\mathbf{f}_1&\mathbf{f}_2\end{pmatrix}$ , prove that $\mathbf{F}^\top\mathbf{F}=\mathbf{I}_3$ and $|\det\mathbf{F}|=1$ .
Deduce that the volume is $|\det\mathbf{M}|$ .
For the upper-left $k\times k$ block $\mathbf{G}_k$ of the Gram matrix ( $k=1,2,3$ ), explain why its determinant is the square of a length, an area, and a volume, respectively.

Hint. Reuse the first two orthonormal directions from the previous exercise, then remove the two projections of the third vector.

# ============================================================
print_header("Exercise 7-32 | Volume in Space")
# ============================================================

# --- Step 1 ---
print_step(1, "Check the three orthonormal directions and the factorization with positive diagonal")
M=np.array([[1,2,1],[2,0,3],[2,2,-2]])
F=np.array([[1,2,2],[2,-2,1],[2,1,-2]])/3
R=np.array([[3,2,1],[0,2,-2],[0,0,3]])
check("F^T F", F.T@F, np.eye(3)); check("FR", F@R, M)
check("three heights", np.diag(F.T@M), [3,2,3])
# --- Step 2 ---
print_step(2, "Check the leading principal minors and the volume")
G=M.T@M
for k,expected in [(1,9),(2,36),(3,324)]:check(f"det(G_{k})", np.linalg.det(G[:k,:k]), expected)
check("volume", abs(np.linalg.det(M)), 18)

============================================================
  Exercise 7-32 | Volume in Space
============================================================

▶ Step 1: Check the three orthonormal directions and the factorization with positive diagonal
----------------------------------------

[F^T F]
  Computed value: [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
  Expected value: [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
  check passed

[FR]
  Computed value: [[ 1.  2.  1.]
 [ 2.  0.  3.]
 [ 2.  2. -2.]]
  Expected value: [[ 1  2  1]
 [ 2  0  3]
 [ 2  2 -2]]
  check passed

[three heights]
  Computed value: [3. 2. 3.]
  Expected value: [3, 2, 3]
  check passed

▶ Step 2: Check the leading principal minors and the volume
----------------------------------------

[det(G_1)]
  Computed value: 9.000000000000002
  Expected value: 9
  check passed

[det(G_2)]
  Computed value: 36.0
  Expected value: 36
  check passed

[det(G_3)]
  Computed value: 323.9999999999999
  Expected value: 324
  check passed

[volume]
  Computed value: 17.999999999999996
  Expected value: 18
  check passed

Exercise 33 (Volume in Higher Dimensions)

Let $1\le k\le n$ , let $\mathbf{v}_0,\ldots,\mathbf{v}_{k-1}\in\mathbb{R}^n$ be linearly independent, and let $\mathbf{M}=\begin{bmatrix}\mathbf{v}_0 & \cdots & \mathbf{v}_{k-1}\end{bmatrix}$ .

Using the earlier strategy of orthogonalization and base times height, prove that $\det(\mathbf{M}^\top\mathbf{M})=\operatorname{Vol}_k(\mathbf{v}_0\wedge\cdots\wedge\mathbf{v}_{k-1})^2$ .
In particular, when $k=n$ , explain why the volume is $|\det\mathbf{M}|$ .

Hint. A rectangular matrix with orthonormal columns still satisfies $\mathbf{F}^\top\mathbf{F}=\mathbf{I}_k$ . (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 34 (Volume in Four Dimensions)

Find the three-dimensional volume of the parallelepiped in $\mathbb{R}^4$ spanned by the three vectors $(1,0,1,0)^\top$ , $(1,1,1,1)^\top$ , and $(-1,1,1,1)^\top$ .

Hint. The original matrix is $4\times3$ ; compute the determinant of the $3\times3$ Gram matrix instead.

# ============================================================
print_header("Exercise 7-34 | Volume in Four Dimensions")
# ============================================================

# --- Step 1 ---
print_step(1, "Build the Gram matrix")
M=np.array([[1,1,-1],[0,1,1],[1,1,1],[0,1,1]])
G=M.T@M
check("G", G, [[2,2,0],[2,4,2],[0,2,4]])
# --- Step 2 ---
print_step(2, "Check the determinant and the three-dimensional volume")
check("det(G)", np.linalg.det(G), 8)
check("volume", np.sqrt(np.linalg.det(G)), 2*np.sqrt(2))

============================================================
  Exercise 7-34 | Volume in Four Dimensions
============================================================

▶ Step 1: Build the Gram matrix
----------------------------------------

[G]
  Computed value: [[2 2 0]
 [2 4 2]
 [0 2 4]]
  Expected value: [[2, 2, 0], [2, 4, 2], [0, 2, 4]]
  check passed

▶ Step 2: Check the determinant and the three-dimensional volume
----------------------------------------

[det(G)]
  Computed value: 7.999999999999998
  Expected value: 8
  check passed

[volume]
  Computed value: 2.82842712474619
  Expected value: 2.8284271247461903
  check passed

Generalized Cross Products¶

Exercise 35 (Generalized Cross Product)

Let $n\ge2$ and take $\mathbf{v}_0,\ldots,\mathbf{v}_{n-2}\in\mathbb{R}^n$ . For any $\mathbf{w}$ , let $\mathbf{M}=\begin{bmatrix}\mathbf{v}_0 & \cdots & \mathbf{v}_{n-2} & \mathbf{w}\end{bmatrix}$ . Take the cofactors of the last column, $C_i=C_{i,n-1}$ , and define the generalized cross product $\mathbf{c}=(C_0,\ldots,C_{n-1})^\top$ ; it does not depend on $\mathbf{w}$ .

When $n=3$ , is this the usual cross product?
Prove that exchanging any two input vectors changes the sign, and that the cross product is zero if two inputs are equal.
Prove that it is linear in each input vector separately.
Prove that $\mathbf{w}^\top\mathbf{c}=\det\begin{bmatrix}\mathbf{v}_0 & \cdots & \mathbf{v}_{n-2} & \mathbf{w}\end{bmatrix}$ .
Prove that $\mathbf{c}$ is perpendicular to every input vector, and hence to their span.

Erratum in the original manuscript. In the determinant of item (d) in the source, the subscript of the second input vector is missing; it should be $\mathbf{v}_2$ in the notation of the original manuscript, which is $\mathbf{v}_1$ in this book’s 0-based notation.

Hint. A cofactor of the last column is just the signed determinant of the smaller matrix obtained by deleting the last column. (This is a derivation/proof exercise; no Python verification is attached.)

Exercise 36 (Cross Product and Volume)

Let $\mathbf{L}=\begin{bmatrix}\mathbf{v}_0 & \cdots & \mathbf{v}_{n-2}\end{bmatrix}\in\mathbb{R}^{n\times(n-1)}$ and $\mathbf{M}=\begin{bmatrix}\mathbf{L} & \mathbf{w}\end{bmatrix}$ , and let $\mathbf{c}=(C_0,\ldots,C_{n-1})^\top$ be the generalized cross product of the previous exercise. Prove in two ways that

\det(\mathbf{L}^\top\mathbf{L})=\operatorname{Vol}_{n-1}^2=\|\mathbf{c}\|^2=\sum_{i=0}^{n-1}C_i^2.

(37)

Choose $\mathbf{w}=\mathbf{c}$ and compare $\det\mathbf{M}$ with $\det(\mathbf{M}^\top\mathbf{M})$ .
Assume without loss of generality that $\mathbf{M}$ is invertible; use the definition of the inverse matrix to find the last row of $\mathbf{M}^{-1}$ and the lower-right entry of $(\mathbf{M}^\top\mathbf{M})^{-1}$ , and use these to give a proof; then also handle the dependent case.

Hint. First handle the case of full column rank, and treat the degenerate case separately, to avoid dividing by zero. (This is a derivation/proof exercise; no Python verification is attached.)

Solution to Exercise 36

If the $n-1$ columns of $\mathbf{L}$ are linearly independent, at least one minor of order $(n-1)$ is nonzero, so $\mathbf{c}\ne0$ . Choosing $\mathbf{w}=\mathbf{c}$ , the previous exercise gives $\det\mathbf{M}=\mathbf{c}^\top\mathbf{c}=\|\mathbf{c}\|^2$ . Moreover $\mathbf{L}^\top\mathbf{c}=\mathbf{0}$ , so
$\mathbf{M}^\top\mathbf{M}=\begin{pmatrix}\mathbf{L}^\top\mathbf{L}&\mathbf{0}\\\mathbf{0}^\top&\|\mathbf{c}\|^2\end{pmatrix}.$
(38)
Comparing determinants gives $\|\mathbf{c}\|^4=\det(\mathbf{L}^\top\mathbf{L})\|\mathbf{c}\|^2$ ; since $\|\mathbf{c}\|^2>0$ , it can be cancelled, which completes the first proof.
For the second method, take any $\mathbf{w}$ that makes $\mathbf{M}$ invertible. When the row vector $\mathbf{c}^\top/\det\mathbf{M}$ is multiplied on the right by $\mathbf{M}$ , the first $n-1$ components are zero and the last is $\mathbf{c}^\top\mathbf{w}/\det\mathbf{M}=1$ , so by the definition of the inverse matrix it is the last row of $\mathbf{M}^{-1}$ . Therefore
$[(\mathbf{M}^\top\mathbf{M})^{-1}]_{n-1,n-1}=[\mathbf{M}^{-1}\mathbf{M}^{-\top}]_{n-1,n-1}=\frac{\|\mathbf{c}\|^2}{(\det\mathbf{M})^2}.$
(39)
On the other hand, write $\mathbf{G}=\mathbf{M}^\top\mathbf{M}=\begin{pmatrix}\mathbf{H}&\mathbf{g}\\\mathbf{g}^\top&s\end{pmatrix}$ , where $\mathbf{H}=\mathbf{L}^\top\mathbf{L}$ is invertible. By the definition of the inverse matrix, its last column $\binom{\mathbf{z}}{\beta}$ satisfies $\mathbf{H}\mathbf{z}+\mathbf{g}\beta=\mathbf{0}$ and $\mathbf{g}^\top\mathbf{z}+s\beta=1$ , so $\beta=(s-\mathbf{g}^\top\mathbf{H}^{-1}\mathbf{g})^{-1}$ . Block elimination also gives $\det\mathbf{G}=\det\mathbf{H}(s-\mathbf{g}^\top\mathbf{H}^{-1}\mathbf{g})$ , so this lower-right entry is $\det\mathbf{H}/\det\mathbf{G}=\det(\mathbf{L}^\top\mathbf{L})/(\det\mathbf{M})^2$ . Comparing the two expressions gives the conclusion.
If the input vectors are dependent, all the maximal minors are zero, so $\mathbf{c}=\mathbf{0}$ ; the Gram matrix is singular, so its determinant is also zero, and the parallelepiped is degenerate with volume zero. Hence the identity also holds in the degenerate case. The identity between the squared volume and the Gram determinant follows from the earlier orthogonalization volume formula (in the independent case) together with this extension to the zero case. $\square$

✓ All exercises completed.