3.4.2 Practice set: determinants

Determinants

Determinants as volume scaling (lesson 3.4.1)

0 of 14 correct

1.
Find $\det \begin{pmatrix} 5 & 2 \\ 3 & 4 \end{pmatrix}$
2.
A map sends $\mathbf{e}_1$ to $(1, 2)$ and $\mathbf{e}_2$ to $(3, 1)$ . Find its determinant
3.
$A = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}$ maps a square of side 2. What is the area of its image?
4.
For which positive $k$ is $\begin{pmatrix} k & 2 \\ 8 & k \end{pmatrix}$ singular?
5.
Find $\det \begin{pmatrix} 3 & 1 & 0 \\ 2 & 4 & 1 \\ 1 & 0 & 2 \end{pmatrix}$
6.
Find $\det \begin{pmatrix} 1 & 4 & 2 \\ 0 & 3 & 0 \\ 2 & 5 & 1 \end{pmatrix}$ , expanding along the row with the most zeros
7.
A box has edges $(2, 1, 0)$ , $(0, 3, 1)$ and $(0, 0, 5)$ . What is its volume?
8.
Find $\det \begin{pmatrix} 2 & 2 & 3 & 4 \\ 0 & -1 & 7 & 8 \\ 0 & 0 & 3 & 12 \\ 0 & 0 & 0 & 0.5 \end{pmatrix}$
9.
$A$ and $B$ are $3 \times 3$ , with $\det A = 3$ and $\det B = 2$ . Find $\det(A B A^\top)$
10.
$\det A = 4$ and $\det B = 6$ . Find $\det(A^{-1}B)$
11.
$\det A = 6$ . Swap rows 1 and 2 of $A$ , then multiply row 3 by 2. What is the new determinant?
12.
$A$ is $3 \times 3$ with $\det A = 5$ . Find $\det(-A)$
13.
$A$ is $2 \times 2$ with $\det A = 2$ . Find $\det(3A)$
14.
NumPy gives $6.7 \times 10^{-16}$ as the determinant of $\begin{pmatrix} 1 & 0 & 1 \\ 4 & 1 & 5 \\ 5 & 2 & 7 \end{pmatrix}$ . What is its exact determinant?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

A determinant by cofactor expansion

Write det(A), the determinant of a square matrix A given as a list of rows, each a list of numbers. Expand along row 1, as in lesson 3.4.1:

A $1 \times 1$ matrix [[a]] has determinant a.
Otherwise, for each column $j$ , delete row 1 and column $j$ , and find the determinant of what is left with det itself. Add up the entries of row 1 times these minors, with signs $+, -, +, -, \ldots$
Raise ValueError if A is empty or not square.

The tests use the matrices of lesson 3.4.1, a row swap, a $4 \times 4$ matrix and a triangular one. With whole-number entries your answer is exact: for lesson 3.4.1's $S$ it must be exactly 0, where NumPy prints $7.5 \times 10^{-15}$ .

The work grows very fast with the size: a $10 \times 10$ matrix has 3,628,800 terms. That is why NumPy eliminates instead.

See the tests

import numpy as np


def test_two_by_two():
    """det([[3, 1], [1, 2]]) is 3·2 − 1·1 = 5"""
    expect_equal(det([[3, 1], [1, 2]]), 5, what="det([[3, 1], [1, 2]])")


def test_one_by_one():
    """A 1 × 1 matrix [[7]] has determinant 7"""
    expect_equal(det([[7]]), 7, what="det([[7]])")


def test_three_by_three():
    """Lesson 3.4.1's C = [[2, 1, 3], [0, 4, 1], [1, 0, 2]] has determinant 5"""
    expect_equal(det([[2, 1, 3], [0, 4, 1], [1, 0, 2]]), 5, what="det(C)")


def test_box():
    """Lesson 3.4.1's box B = [[2, 1, 1], [0, 2, 1], [0, 0, 2]] has volume 8"""
    expect_equal(det([[2, 1, 1], [0, 2, 1], [0, 0, 2]]), 8, what="det(B)")


def test_row_swap():
    """Swapping rows 1 and 3 of C flips the sign: −5"""
    expect_equal(det([[1, 0, 2], [0, 4, 1], [2, 1, 3]]), -5, what="det(C with rows 1 and 3 swapped)")


def test_singular_exact():
    """Lesson 3.4.1's S = [[1, 3, 4], [3, 1, 4], [1, 6, 7]] gives exactly 0"""
    expect_equal(det([[1, 3, 4], [3, 1, 4], [1, 6, 7]]), 0, what="det(S), which NumPy prints as 7.5e-15")


def test_four_by_four():
    """A 4 × 4 matrix matches NumPy: [[1, 2, 0, 1], [0, 3, 1, 2], [2, 0, 1, 1], [1, 1, 0, 3]] gives 17"""
    A = [[1, 2, 0, 1], [0, 3, 1, 2], [2, 0, 1, 1], [1, 1, 0, 3]]
    expect_close(det(A), float(np.linalg.det(np.array(A, dtype=float))), what="det of the 4 × 4 matrix", rel=1e-9)
    expect_equal(det(A), 17, what="det of the 4 × 4 matrix")


def test_triangular():
    """A triangular 4 × 4 matrix with diagonal 2, −1, 3, 0.5 gives their product, −3"""
    A = [[2, 2, 3, 4], [0, -1, 7, 8], [0, 0, 3, 12], [0, 0, 0, 0.5]]
    expect_close(det(A), -3.0, what="det of the triangular matrix")


def test_not_square():
    """A 2 × 3 matrix has no determinant: ValueError"""
    try:
        det([[1, 2, 3], [4, 5, 6]])
    except ValueError:
        return
    raise AssertionError("expected ValueError for a 2 × 3 matrix")


def test_ragged():
    """Rows of different lengths, [[1, 2], [3]], are not a square matrix: ValueError"""
    try:
        det([[1, 2], [3]])
    except ValueError:
        return
    except IndexError:
        raise AssertionError("expected ValueError for rows of different lengths, got IndexError")
    raise AssertionError("expected ValueError for rows of different lengths")


def test_empty():
    """An empty matrix [] has no determinant: ValueError"""
    try:
        det([])
    except ValueError:
        return
    raise AssertionError("expected ValueError for an empty matrix")

Determinants

Coding exercise

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