How to Add and Subtract Matrices? (+FREE Worksheet!)
Adding and Subtracting Matrices – Example 1:
\(\begin{bmatrix} 2 & -5 & -3 \end{bmatrix}+ \begin{bmatrix}1 & -2 & -3 \end{bmatrix}\)
Add and Subtract Matrices: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Check dimensionsRows by columns determines what operation is legal.
- Use the correct ruleAddition is entry-by-entry; multiplication is row-by-column.
- Interpret the resultFor systems, translate the matrix answer back into variables.
Worked examples
Add matrices
- The matrices have the same size.
- Add matching entries.
- Compute each position.
Multiplication size
- Inner dimensions match: 3 and 3.
- The product is allowed.
- Outer dimensions give the result size.
Try one before moving on
Add and Subtract Matrices: pop-up practice
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: \(\begin{bmatrix}-1-5 & 2-(-2) \\3-6 & 4-1 \\0-(-3) & -2-4\end{bmatrix}= \begin{bmatrix}-6 & 4 \\-3 & 3 \\3 & -6\end{bmatrix}\)
Exercises for Adding and Subtracting Matrices
- \(\color{blue}{\begin{bmatrix}6 & 8 \\-14 & 33 \end{bmatrix}-\begin{bmatrix}12 & 5 \\-27 & -8 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}12 & 21 \\-17 & 33 \end{bmatrix}-\begin{bmatrix}5& -8 \\2 & 19 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}-5 & 2& -2 \\4 & -2&0 \end{bmatrix}-\begin{bmatrix}6 & -5&-6 \\1 & 3&-3 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}-6r+t \\-r\\6s \end{bmatrix}+\begin{bmatrix}6r \\-4t\\-3r+2 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}16 & -4 \\-38 & 24 \end{bmatrix}+\begin{bmatrix}9 & -6 \\5& 2 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}18 & -5 \\-32 & 14 \end{bmatrix}+\begin{bmatrix}11& -6 \\7 & 2 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}-4n & n+m \\-2n & -4m\end{bmatrix}+\begin{bmatrix}4 & -5 \\3 & 0 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}z-5 \\-6\\-1-6z\\3y \end{bmatrix}+\begin{bmatrix}-3y\\3z\\5+z\\4z \end{bmatrix}}\)
Solve.
- \(\color{blue}{\begin{bmatrix}-6 & 3 \\13 & 41 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}7 & 29 \\-19 & 14 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}-11 & 7&4 \\3 & -5&3 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}t \\-r-4t\\6s-3r+2 \end{bmatrix}} \)
- \(\color{blue}{\begin{bmatrix}25 & -10 \\-33 & 26 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}29 & -11 \\-25 & 16 \end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}-4n+4 & n+m-5 \\-2n+3 & -4m\end{bmatrix}}\)
- \(\color{blue}{\begin{bmatrix}z-5-3y \\-6+3z\\4-5z\\3y+4z \end{bmatrix}} \)
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