Practice Matrix Addition, Subtraction, and Scalar 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

Matrix addition and subtraction are performed entry by entry on matrices of the same dimensions. Scalar multiplication multiplies every entry of a matrix by a single number (the scalar).

Adding matrices is like adding two spreadsheets cell by cell. If spreadsheet $A$ has sales for January and $B$ has sales for February, then $A + B$ gives total sales in each cell. Scalar multiplication is like giving everyone in the spreadsheet a 10% raise—multiply every entry by 1.1.

Showing a random 20 of 50 problems.

Example 1

easy

Compute

2 \begin{pmatrix} 3 \\ -1 \end{pmatrix}

Example 2

hard

Let

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

. Find a matrix

B

such that

A+B=2A

Example 3

easy

Compute

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

Example 4

easy

Compute

\begin{pmatrix} 5 & 7 \\ 9 & 2 \end{pmatrix} - \begin{pmatrix} 1 & 3 \\ 4 & 0 \end{pmatrix}

Example 5

easy

Compute

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

Example 6

challenge

For what matrices

A

does

A+A^T

have all entries doubled compared to the diagonal of

A

Example 7

medium

Show that

A+(B+C)=(A+B)+C

for

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

B=\begin{bmatrix}2&3\\4&5\end{bmatrix}

C=\begin{bmatrix}1&1\\1&1\end{bmatrix}

Example 8

medium

Find

x

\begin{pmatrix} x & 1 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} 4 & 0 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 9 & 1 \\ 3 & 5 \end{pmatrix}

Example 9

easy

Compute

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

Example 10

easy

Compute

\begin{bmatrix} 2 & 3 \\ 5 & 1 \end{bmatrix} + \begin{bmatrix} 4 & -1 \\ 0 & 2 \end{bmatrix}

Example 11

easy

Compute

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

Example 12

challenge

A + B = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}

and

A - B = \begin{pmatrix} 1 & 3 \\ -1 & 1 \end{pmatrix}

, find

A

Example 13

medium

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

and

B=\begin{bmatrix} 1 & 4 \\ -2 & 1 \end{bmatrix}

, find

2A+3B

Example 14

medium

Compute

\tfrac{1}{3}\begin{bmatrix} 6 & 9 \\ -3 & 12 \end{bmatrix}-\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}

Example 15

medium

Compute

\begin{pmatrix} 4 & 2 \\ 6 & 8 \end{pmatrix} - 2\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}

Example 16

easy

Compute

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

Example 17

easy

Compute

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

Example 18

hard

Find

a

and

b

such that

a\begin{bmatrix}1\\2\end{bmatrix}+b\begin{bmatrix}3\\-1\end{bmatrix}=\begin{bmatrix}5\\3\end{bmatrix}

Example 19

medium

Let

A

be a

3\times 4

matrix. What dimensions must

B

have so that

A+B

is defined?

Example 20

medium

Find the unknown entry

y

\begin{bmatrix} 3 & y \\ 2 & 1 \end{bmatrix}+\begin{bmatrix} 4 & 2 \\ y & 5 \end{bmatrix}=\begin{bmatrix} 7 & 9 \\ 8 & 6 \end{bmatrix}

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Related Concepts

Matrix Definition Matrix Multiplication Determinant