Practice Matrix Multiplication in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Multiplying matrices $A$ ( $m \times n$ ) and $B$ ( $n \times p$ ) by taking dot products of rows of $A$ with columns of $B$ to produce an $m \times p$ result.

Imagine each row of $A$ as a question and each column of $B$ as an answer key. You 'grade' each row against each column by multiplying corresponding entries and summing. This is why column count of $A$ must match row count of $B$ —the question and answer key must have the same length.

Showing a random 20 of 50 problems.

Example 1

hard

Find

2\times 2

matrices

A,B

with

AB = 0

but

A,B \ne 0

Example 2

easy

Compute

\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \end{pmatrix}

Example 3

challenge

For

A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}

and any

2 \times 2

matrix

B

, why does

AB = BA

? Compute

AB

for

B = \begin{pmatrix} 1 & 3 \\ 5 & 7 \end{pmatrix}

Example 4

medium

Compute

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and explain.

Example 5

easy

True/false: matrix multiplication is associative —

(AB)C = A(BC)

whenever defined.

Example 6

easy

Can you multiply a

2 \times 3

matrix by a

2 \times 3

matrix?

Example 7

medium

Compute

\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 1 & -1 \end{pmatrix}

Example 8

hard

Find

A^2

for

A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}

Example 9

medium

Compute

\begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix}\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}

Example 10

medium

Compute

\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}^2

Example 11

challenge

Find a nonzero

2 \times 2

matrix

A

with

A^2 = 0

(the zero matrix). Give one example and verify.

Example 12

easy

Can you multiply a

2 \times 3

matrix by a

2 \times 3

matrix? Why or why not?

Example 13

easy

Compute the dot product needed for

(AB)_{11}

when

A

's first row is

(2,-1,3)

and

B

's first column is

(4,5,-2)^T

Example 14

easy

Compute

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix}

Example 15

medium

For

A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and

B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

, compute both

AB

and

BA

, showing they differ.

Example 16

medium

For

A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

, compute

A^2

Example 17

easy

Compute the dot product for the product entry: row

\begin{pmatrix} 1 & 2 \end{pmatrix}

times column

\begin{pmatrix} 3 \\ 4 \end{pmatrix}

Example 18

medium

For the rotation matrix

R(\theta)=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

, compute

R(\theta)R(\phi)

Example 19

hard

Compute

\begin{bmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 4 \\ -1 \end{bmatrix}

Example 20

easy

Compute

\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 7 \\ -3 \end{pmatrix}

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Related Concepts

Matrix Addition, Subtraction, and Scalar Multiplication Matrix Definition Determinant Inverse Matrix Solving Systems of Equations with Matrices