Matrix Addition, Subtraction, and Scalar Multiplication Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Matrix Addition, Subtraction, and Scalar 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

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.

Read the full concept explanation →

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: Add or subtract matrices entry by entry (same dimensions required), and scalar multiplication scales every entry.

Common stuck point: The procedure for matrix addition, subtraction, and scalar multiplication is the easy part; the trap is adding mismatched-size matrices. Asking "For $+/-$ , do the two matrices have exactly the same dimensions?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: For $+/-$ , do the two matrices have exactly the same dimensions?

Worked Examples

Example 1

easy

Compute

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

Answer

\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

First step

Step 1: Add corresponding entries:

a_{ij} + b_{ij}

Full solution

2
Step 2: $\begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$ .
3
Check: Each entry is the sum of the entries in the same position ✓

Matrix addition adds corresponding entries element by element. Both matrices must have the same dimensions for addition to be defined.

Example 2

medium

Compute

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

Example 3

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 4

medium

Given

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

and

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

, compute

4A-3B

Example 5

hard

Demonstrate that

(k+m)A=kA+mA

for

k=2

m=3

, and

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

Practice Problems

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

Example 1

easy

Compute

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

Example 2

medium

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

and

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

, find

A - 2B

Example 3

easy

Compute

X

X + \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}

Example 14

medium

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

, compute

A + A + A

and confirm it equals

3A

Example 15

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 16

medium

Compute

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

Example 17

medium

Compute

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

Example 18

medium

Find

Y

3Y = \begin{pmatrix} 6 & 9 \\ 12 & 0 \end{pmatrix}

Example 19

medium

Compute

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

Example 20

challenge

Find matrices satisfying

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

Example 21

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 22

challenge

For what scalar

medium

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

and

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

, find

2A+3B

Example 29

medium

Solve for

X

X-\begin{bmatrix} 2 & 5 \\ 1 & 0 \end{bmatrix}=\begin{bmatrix} 4 & 1 \\ 3 & 6 \end{bmatrix}

Example 30

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}

Example 31

medium

Compute

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

Example 32

medium

Find scalar

k

such that

k\begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix}=\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

Example 33

medium

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

, compute

A-A

Example 34

hard

Given

A+B=\begin{bmatrix}10&4\\6&8\end{bmatrix}

and

2A-B=\begin{bmatrix}2&5\\6&4\end{bmatrix}

, find

A

and

B

Example 35

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 36

hard

Determine all values of

t

for which

\begin{bmatrix} t & 1 \\ 2 & t \end{bmatrix}+\begin{bmatrix} 1 & t \\ t & 1 \end{bmatrix}

equals

\begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix}

Example 37

hard

Let

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

. Find a matrix

B

such that

A+B=2A

Example 38

medium

Compute the result of

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

Example 39

easy

Compute

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

Example 40

challenge

For what matrices

A

does

A+A^T

have all entries doubled compared to the diagonal of

A

Example 41

challenge

Find scalars

\alpha,\beta,\gamma

such that

\alpha\begin{bmatrix}1\\0\end{bmatrix}+\beta\begin{bmatrix}0\\1\end{bmatrix}+\gamma\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}5\\3\end{bmatrix}

with

\gamma=2

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Related Concepts

Matrix Definition Matrix Multiplication Determinant

Background Knowledge

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

matrix definition