Matrix Multiplication Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Matrix Multiplication.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Each result entry is the dot product of a row of $A$ with a column of $B$ , and inner dimensions must match.

Common stuck point: The procedure for matrix multiplication is the easy part; the trap is multiplying when inner dimensions disagree. Asking "Does the column count of $A$ equal the row count of $B$ , and am I dotting rows with columns?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the column count of $A$ equal the row count of $B$ , and am I dotting rows with columns?

Worked Examples

Example 1

medium

Compute

\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}

Answer

\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

First step

Step 1: Entry

(1,1)

1 \cdot 5 + 2 \cdot 7 = 5 + 14 = 19

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Example 2

hard

Compute

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

Example 3

challenge

Use the Fibonacci matrix identity

F^n=\begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}

and

F^{m+n}=F^m F^n

to derive an identity relating

F_{m+n}

F_m, F_{m-1}, F_n, F_{n+1}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Can you multiply a

2 \times 3

matrix by a

2 \times 3

matrix? Why or why not?

Example 2

medium

Compute

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & -2 \\ 5 & 7 \end{bmatrix}

Example 3

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 4

easy

What are the dimensions of

AB

A

2 \times 3

and

B

3 \times 2

Example 5

easy

Compute

\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}

Example 6

easy

Compute

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

Example 7

easy

Can you multiply a

2 \times 3

matrix by a

2 \times 3

matrix?

Example 8

easy

Compute

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

Example 9

easy

Compute the (1,1) entry of

\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 4 & 5 \\ 6 & 7 \end{pmatrix}

Example 10

easy

Is matrix multiplication commutative in general?

Example 11

medium

Compute

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}

Example 12

medium

Compute

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

Example 13

medium

Show that

AB \neq BA

for

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

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

. Compute

AB

Example 14

medium

Compute

\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}\begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}

Example 15

medium

Compute

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

Example 16

medium

Compute

A^2

for

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

Example 17

medium

Compute

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

Example 18

medium

Compute

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

Example 19

medium

Compute the (2,1) entry of

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

Example 20

challenge

Find a nonzero

2 \times 2

matrix

A

with

A^2 = 0

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

Example 21

challenge

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

, find

A^3

Example 22

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 23

easy

Can a

3 \times 4

matrix be multiplied (on the right) by a

5 \times 2

matrix?

Example 24

easy

Compute

\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}

Example 25

easy

For

A

3\times 3

matrix, what are the dimensions of

A^4

Example 26

easy

True/false: matrix multiplication is associative —

(AB)C = A(BC)

whenever defined.

Example 27

medium

Compute

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

Example 28

medium

Compute

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

Example 29

medium

For

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

, compute

A^2

Example 30

medium

Compute

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

and explain the geometric effect.

Example 31

medium

Compute

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

and explain.

Example 32

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 33

medium

Show that

(AB)^T = B^T A^T

by computing both sides for

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

B=\begin{pmatrix} 5 & 0 \\ 0 & 6 \end{pmatrix}

Example 34

medium

Compute

\begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}

Example 35

medium

Compute

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

Example 36

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 37

hard

For

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

, find a closed form for

A^n

Example 38

hard

Find

A^2

for

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

Example 39

hard

Find a

2\times 2

matrix

A

with

A^2 = -I

Example 40

hard

Find

2\times 2

matrices

A,B

with

AB = 0

but

A,B \ne 0

Example 41

hard

Compute

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

Example 42

hard

For the projection matrix

P = \tfrac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}

, compute

P^2

and explain.

Example 43

challenge

For Fibonacci matrix

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

, compute

F^3

and identify the entries.

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Related Concepts

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

Background Knowledge

These ideas may be useful before you work through the harder examples.

matrix operationsmatrix definition