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 AA has sales for January and BB has sales for February, then A+BA + 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(3−1)2 \begin{pmatrix} 3 \\ -1 \end{pmatrix}.

Example 2

hard
Let A=[1234]A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}. Find a matrix BB such that A+B=2AA+B=2A.

Example 3

easy
Compute 5[1−203]5\begin{bmatrix} 1 & -2 \\ 0 & 3 \end{bmatrix}.

Example 4

easy
Compute (5792)−(1340)\begin{pmatrix} 5 & 7 \\ 9 & 2 \end{pmatrix} - \begin{pmatrix} 1 & 3 \\ 4 & 0 \end{pmatrix}.

Example 5

easy
Compute [123456]+[011022]\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 AA does A+ATA+A^T have all entries doubled compared to the diagonal of AA?

Example 7

medium
Show that A+(B+C)=(A+B)+CA+(B+C)=(A+B)+C for A=[1001]A=\begin{bmatrix}1&0\\0&1\end{bmatrix}, B=[2345]B=\begin{bmatrix}2&3\\4&5\end{bmatrix}, C=[1111]C=\begin{bmatrix}1&1\\1&1\end{bmatrix}.

Example 8

medium
Find xx if (x123)+(4012)=(9135)\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 (1234)+(5678)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}.

Example 10

easy
Compute [2351]+[4−102]\begin{bmatrix} 2 & 3 \\ 5 & 1 \end{bmatrix} + \begin{bmatrix} 4 & -1 \\ 0 & 2 \end{bmatrix}.

Example 11

easy
Compute −3[2−14]-3\begin{bmatrix} 2 \\ -1 \\ 4 \end{bmatrix}.

Example 12

challenge
If A+B=(5555)A + B = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix} and A−B=(13−11)A - B = \begin{pmatrix} 1 & 3 \\ -1 & 1 \end{pmatrix}, find AA.

Example 13

medium
If A=[2−103]A=\begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix} and B=[14−21]B=\begin{bmatrix} 1 & 4 \\ -2 & 1 \end{bmatrix}, find 2A+3B2A+3B.

Example 14

medium
Compute 13[69−312]−[1203]\tfrac{1}{3}\begin{bmatrix} 6 & 9 \\ -3 & 12 \end{bmatrix}-\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}.

Example 15

medium
Compute (4268)−2(1111)\begin{pmatrix} 4 & 2 \\ 6 & 8 \end{pmatrix} - 2\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}.

Example 16

easy
Compute (2013)+(0000)\begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}.

Example 17

easy
Compute 3(1204)3 \begin{pmatrix} 1 & 2 \\ 0 & 4 \end{pmatrix}.

Example 18

hard
Find aa and bb such that a[12]+b[3−1]=[53]a\begin{bmatrix}1\\2\end{bmatrix}+b\begin{bmatrix}3\\-1\end{bmatrix}=\begin{bmatrix}5\\3\end{bmatrix}.

Example 19

medium
Let AA be a 3×43\times 4 matrix. What dimensions must BB have so that A+BA+B is defined?

Example 20

medium
Find the unknown entry yy if [3y21]+[42y5]=[7986]\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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