3.2.4 Practice set: matrices

Matrices and their products

Matrices and matrix multiplication (lesson 3.2.1)

0 of 6 correct

1.
For $A = \begin{pmatrix} 4 & -2 & 7 \\ 1 & 0 & 5 \\ 3 & 8 & -6 \end{pmatrix}$ , what is $a_{32}$ ?
2.
For $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ and $\mathbf{x} = (1, 0, -1)$ , find the second entry of $A\mathbf{x}$
3.
$A$ has columns $(1, 2)$ and $(4, -1)$ . Find the second entry of $A\mathbf{x}$ for $\mathbf{x} = (3, -2)$
4.
For $A = \begin{pmatrix} 2 & 0 & 1 \\ 1 & 3 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 \\ 0 & 1 \\ 4 & -1 \end{pmatrix}$ , find the entry in row 2, column 2 of $AB$
5.
$A$ is $2 \times 3$ and $B$ is $3 \times 2$ . How many entries does $BA$ have?
6.
For $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ , multiplying entry by entry gives a top-right entry of 2. What is the top-right entry of the matrix product $AB$ ?

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.

Matrices as maps

Matrices as linear maps (lesson 3.2.2)

0 of 7 correct

1.
A linear map sends $\mathbf{e}_1$ to $(2, 5)$ and $\mathbf{e}_2$ to $(-1, 3)$ . Find the first entry of the image of $(4, 2)$
2.
The shear $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$ slides each point up by twice its first component. Where does $(3, 1)$ go? Give the second entry
3.
Rotate $(2, 0)$ through 45° anticlockwise. What is the second entry of the image? (type sqrt for a square root)
4.
Reflect $(5, 4)$ in the $x_2$ -axis, then project the result onto the $x_1$ -axis. What is the first entry?
5.
$S = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$ stretches and $H = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ shears. Stretch first, then shear: find the entry in row 1, column 2 of the single matrix that does both
6.
For $T(\mathbf{x}) = 2\mathbf{x} + (1, 1)$ , find the first entry of $T(\mathbf{u} + \mathbf{v}) - T(\mathbf{u}) - T(\mathbf{v})$ , for any $\mathbf{u}$ and $\mathbf{v}$
7.
A linear map sends $\mathbf{e}_1$ to $(1, 4)$ and $\mathbf{e}_2$ to $(2, 3)$ . What is the entry $a_{21}$ of its matrix?

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.

Transpose, identity and inverse

Transpose, identity and inverse (lesson 3.2.3)

0 of 8 correct

1.
For $A = \begin{pmatrix} 2 & 9 \\ -4 & 1 \end{pmatrix}$ , what is the entry in row 2, column 1 of $A^\top$ ?
2.
For $\mathbf{u} = (1, 2, 3)$ and $\mathbf{v} = (-2, 0, 5)$ , find $\mathbf{u}^\top\mathbf{v}$
3.
For $\mathbf{u} = (2, -1, 2)$ , find $\lVert \mathbf{u} \rVert$ from $\mathbf{u}^\top\mathbf{u}$
4.
Find the entry in row 2, column 1 of the inverse of $\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$
5.
Find the entry in row 1, column 1 of the inverse of $\begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix}$
6.
Solve $\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\mathbf{x} = (5, 9)$ . What is $x_2$ ?
7.
For which positive $t$ does $\begin{pmatrix} t & 3 \\ 2 & t - 1 \end{pmatrix}$ have no inverse?
8.
For $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ , find the entry in row 2, column 2 of $(AB)^\top$

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

Matrix multiplication and the 2 × 2 inverse

Write two functions. Matrices are lists of rows, so $\begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$ is [[2, 1], [-1, 3]].

matmul(A, B) returns $AB$ as a list of rows, built from the rule of lesson 3.2.1: entry $(i, j)$ is row $i$ of A dotted with column $j$ of B. Do not use NumPy's @. Raise ValueError if the rows of A are not as long as the columns of B.
inverse_2x2(A) returns the inverse of a $2 \times 2$ matrix by the formula of lesson 3.2.3. Raise ValueError if $ad - bc = 0$ .

The tests try square and rectangular matrices, both orders of the same pair, a row times a column, and a pair whose shapes do not match. For the inverse they try whole numbers, decimals, a rotation and two matrices with no inverse.

See the tests

import math

import numpy as np


def test_square():
    """[[2, 1], [−1, 3]] times [[1, 0], [2, 1]] is [[4, 1], [5, 3]]"""
    expect_close(matmul([[2, 1], [-1, 3]], [[1, 0], [2, 1]]), [[4, 1], [5, 3]], what="matmul(A, C)")


def test_order_matters():
    """The same two matrices in the other order give [[2, 1], [3, 5]]"""
    expect_close(matmul([[1, 0], [2, 1]], [[2, 1], [-1, 3]]), [[2, 1], [3, 5]], what="matmul(C, A)")


def test_rectangular():
    """A 2 × 3 matrix times a 3 × 2 matrix gives a 2 × 2 matrix"""
    expect_close(matmul([[1, 2, 0], [3, -1, 4]], [[1, 0], [2, 1], [0, 3]]), [[5, 2], [1, 11]], what="matmul(M, N)")


def test_other_shape():
    """The same pair the other way round gives a 3 × 3 matrix"""
    expected = [[1, 2, 0], [5, 3, 4], [9, -3, 12]]
    expect_close(matmul([[1, 0], [2, 1], [0, 3]], [[1, 2, 0], [3, -1, 4]]), expected, what="matmul(N, M)")


def test_column():
    """A matrix times a column: 6 chairs and 2 tables need 32 legs, 72 screws and 12 boards"""
    expect_close(matmul([[4, 4], [8, 12], [1, 3]], [[6], [2]]), [[32], [72], [12]], what="matmul(P, [[6], [2]])")


def test_row_times_column():
    """A row times a column is a 1 × 1 matrix, the dot product: [[5]]"""
    expect_close(matmul([[3, 1]], [[1], [2]]), [[5]], what="matmul([[3, 1]], [[1], [2]])")


def test_matches_numpy():
    """Random 4 × 3 and 3 × 5 matrices agree with NumPy's @"""
    rng = np.random.default_rng(3)
    A = rng.integers(-5, 6, size=(4, 3))
    B = rng.integers(-5, 6, size=(3, 5))
    expect_close(matmul(A.tolist(), B.tolist()), A @ B, what="matmul of a 4 × 3 and a 3 × 5 matrix")


def test_shape_mismatch():
    """Two 2 × 3 matrices cannot be multiplied: ValueError"""
    try:
        matmul([[1, 2, 3], [4, 5, 6]], [[1, 2, 3], [4, 5, 6]])
    except ValueError:
        return
    raise AssertionError("expected ValueError: rows of length 3 cannot meet columns of length 2")


def test_inverse_whole_numbers():
    """The inverse of [[2, 1], [1, 1]] is [[1, −1], [−1, 2]]"""
    expect_close(inverse_2x2([[2, 1], [1, 1]]), [[1, -1], [-1, 2]], what="inverse_2x2([[2, 1], [1, 1]])")


def test_inverse_decimals():
    """The inverse of [[4, 2], [1, 3]] is [[0.3, −0.2], [−0.1, 0.4]]"""
    expect_close(inverse_2x2([[4, 2], [1, 3]]), [[0.3, -0.2], [-0.1, 0.4]], what="inverse_2x2([[4, 2], [1, 3]])")


def test_inverse_undoes():
    """[[2, 1], [−1, 3]] times its inverse is the identity, in both orders"""
    A = np.array([[2, 1], [-1, 3]])
    inv = np.asarray(inverse_2x2([[2, 1], [-1, 3]]), dtype=float)
    expect_close(A @ inv, np.eye(2), what="A times your inverse")
    expect_close(inv @ A, np.eye(2), what="your inverse times A")


def test_rotation():
    """The inverse of the 30° rotation is the rotation by −30°"""
    t = math.radians(30)
    R = [[math.cos(t), -math.sin(t)], [math.sin(t), math.cos(t)]]
    back = [[math.cos(t), math.sin(t)], [-math.sin(t), math.cos(t)]]
    expect_close(inverse_2x2(R), back, what="inverse of the 30° rotation")


def test_singular():
    """[[1, 2], [0.5, 1]] has ad − bc = 0, so it has no inverse: ValueError"""
    try:
        inverse_2x2([[1, 2], [0.5, 1]])
    except ValueError:
        return
    raise AssertionError("expected ValueError: ad − bc = 1 − 2 × 0.5 = 0")


def test_zero_matrix():
    """The zero matrix has no inverse: ValueError"""
    try:
        inverse_2x2([[0, 0], [0, 0]])
    except ValueError:
        return
    raise AssertionError("expected ValueError for the zero matrix")

Matrices and their products

Matrices as maps

Transpose, identity and inverse

Coding exercise

Free further reading