Adding and Subtracting Matrices: Examples and Free Practice

To add or subtract matrices, combine entries in the same row and column. Both matrices must have exactly the same dimensions. Use the worked examples below, try the eight free practice problems, and check your answers.

Worked examples

Adding and Subtracting Matrices, Example 1:

\(\begin{bmatrix} 2 & -5 & -3 \end{bmatrix}+ \begin{bmatrix}1 & -2 & -3 \end{bmatrix}\)

Solution:

Add the elements in the matching positions: \(\begin{bmatrix} 2+1 & -5+(-2) & -3+(-3) \end{bmatrix}= \begin{bmatrix}3 & -7 & -6 \end{bmatrix}\)

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Adding and Subtracting Matrices, Example 2:

\(\begin{bmatrix}3 & 6 \\-1 & -3 \\-5 & -1\end{bmatrix}+ \begin{bmatrix}0 & -1 \\6 & 0 \\2 & 3\end{bmatrix} \)

Solution:

Add the elements in the matching positions: \(\begin{bmatrix}3+0 & 6+(-1) \\-1+6 & -3+0 \\-5+2 & -1+3\end{bmatrix} = \begin{bmatrix}3 & 5 \\5 & -3 \\-3 & 2\end{bmatrix} \)

Adding and Subtracting Matrices, Example 3:

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

Solution:

Add the elements in the matching positions: \(\begin{bmatrix}2+3 & -1+1 \\0+5 & 1+(-2) \end{bmatrix}=\begin{bmatrix}5 & 0 \\5 & -1 \end{bmatrix}\)

Adding and Subtracting Matrices, Example 4:

\(\begin{bmatrix}-1 & 2 \\3 & 4 \\0 & -2\end{bmatrix}- \begin{bmatrix}5 & -2 \\6 & 1 \\-3 & 4\end{bmatrix} \)

Solution:

Subtract the elements in the matching positions: \(\begin{bmatrix}-1-5 & 2-(-2) \\3-6 & 4-1 \\0-(-3) & -2-4\end{bmatrix}= \begin{bmatrix}-6 & 4 \\-3 & 3 \\3 & -6\end{bmatrix}\)

Eight free practice problems

Work these on paper, then compare each entry with the answers below. The existing book-preview button is a separate resource.

1. \(\color{blue}{\begin{bmatrix}6 & 8 \\-14 & 33 \end{bmatrix}-\begin{bmatrix}12 & 5 \\-27 & -8 \end{bmatrix}}\)
2. \(\color{blue}{\begin{bmatrix}12 & 21 \\-17 & 33 \end{bmatrix}-\begin{bmatrix}5& -8 \\2 & 19 \end{bmatrix}}\)
3. \(\color{blue}{\begin{bmatrix}-5 & 2& -2 \\4 & -2&0 \end{bmatrix}-\begin{bmatrix}6 & -5&-6 \\1 & 3&-3 \end{bmatrix}}\)
4. \(\color{blue}{\begin{bmatrix}-6r+t \\-r\\6s \end{bmatrix}+\begin{bmatrix}6r \\-4t\\-3r+2 \end{bmatrix}}\)
5. \(\color{blue}{\begin{bmatrix}16 & -4 \\-38 & 24 \end{bmatrix}+\begin{bmatrix}9 & -6 \\5& 2 \end{bmatrix}}\)
6. \(\color{blue}{\begin{bmatrix}18 & -5 \\-32 & 14 \end{bmatrix}+\begin{bmatrix}11& -6 \\7 & 2 \end{bmatrix}}\)
7. \(\color{blue}{\begin{bmatrix}-4n & n+m \\-2n & -4m\end{bmatrix}+\begin{bmatrix}4 & -5 \\3 & 0 \end{bmatrix}}\)
8. \(\color{blue}{\begin{bmatrix}z-5 \\-6\\-1-6z\\3y \end{bmatrix}+\begin{bmatrix}-3y\\3z\\5+z\\4z \end{bmatrix}}\)

Adding and Subtracting Matrices

Answers to the eight practice problems

1. \(\color{blue}{\begin{bmatrix}-6 & 3 \\13 & 41 \end{bmatrix}}\)
2. \(\color{blue}{\begin{bmatrix}7 & 29 \\-19 & 14 \end{bmatrix}}\)
3. \(\color{blue}{\begin{bmatrix}-11 & 7&4 \\3 & -5&3 \end{bmatrix}}\)
4. \(\color{blue}{\begin{bmatrix}t \\-r-4t\\6s-3r+2 \end{bmatrix}} \)
5. \(\color{blue}{\begin{bmatrix}25 & -10 \\-33 & 26 \end{bmatrix}}\)
6. \(\color{blue}{\begin{bmatrix}29 & -11 \\-25 & 16 \end{bmatrix}}\)
7. \(\color{blue}{\begin{bmatrix}-4n+4 & n+m-5 \\-2n+3 & -4m\end{bmatrix}}\)
8. \(\color{blue}{\begin{bmatrix}z-5-3y \\-6+3z\\4-5z\\3y+4z \end{bmatrix}} \)

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