Matrix Addition and Subtraction
This page follows the JoeCMath video Matrix Addition & Subtraction Under 4 Mins. The video checks when matrix addition or subtraction is allowed, then works one addition example and one subtraction example entry by entry.
Main Idea
The video starts with the condition that matters before any arithmetic happens: the matrices need the same number of rows and the same number of columns.
If
\[A=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \quad\text{and}\quad B=\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix},\]then addition and subtraction are done position by position:
\[A+B= \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}\]and
\[A-B= \begin{bmatrix} a_{11}-b_{11} & a_{12}-b_{12} \\ a_{21}-b_{21} & a_{22}-b_{22} \end{bmatrix}.\]Watch this section: rows and columns need to agree at 0:12.
Rows and Columns Need to Match
The video highlights the dimensions before adding or subtracting. A $2\times 2$ matrix can be added to another $2\times 2$ matrix. A $2\times 3$ matrix can be added to another $2\times 3$ matrix.
But the video also shows a mismatch like
\[\begin{bmatrix} 11 & 5 & 2 \\ 8 & 4 & 17 \end{bmatrix} + \begin{bmatrix} 2 & 6 \\ 3 & 10 \end{bmatrix}.\]The first matrix is $2\times 3$, while the second matrix is $2\times 2$. Since the orders do not match, that addition is not defined.
Watch this section: same number of rows and columns at 0:12.
Matrix Addition Example
The addition example in the video is
\[\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}.\]Both matrices are $2\times 2$, so the answer will also be a $2\times 2$ matrix. Add entries that are in the same position:
\[\begin{aligned} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} &= \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} \\ &= \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}. \end{aligned}\]The video keeps the arithmetic lined up by matching the top-left entry with the top-left entry, the top-right entry with the top-right entry, and so on.
Watch this section: matrix addition example at 0:47.
Matrix Subtraction Example
The subtraction example in the video uses two $2\times 3$ matrices:
\[\begin{bmatrix} 5 & 9 & 12 \\ 4 & 3 & 7 \end{bmatrix} - \begin{bmatrix} 8 & 3 & 6 \\ 7 & 1 & 6 \end{bmatrix}.\]The dimensions match, so subtract corresponding entries:
\[\begin{aligned} \begin{bmatrix} 5 & 9 & 12 \\ 4 & 3 & 7 \end{bmatrix} - \begin{bmatrix} 8 & 3 & 6 \\ 7 & 1 & 6 \end{bmatrix} &= \begin{bmatrix} 5-8 & 9-3 & 12-6 \\ 4-7 & 3-1 & 7-6 \end{bmatrix} \\ &= \begin{bmatrix} -3 & 6 & 6 \\ -3 & 2 & 1 \end{bmatrix}. \end{aligned}\]The negative entries come from subtracting a larger entry from a smaller one in the same position.
Watch this section: matrix subtraction example at 2:10.
Review
At the end of the video, the review keeps the rule compact:
| Question | Answer |
|---|---|
| When can you add or subtract matrices? | When the rows and columns match. |
| What size is the result? | The result has the same dimensions as the original matrices. |
| Which entries combine? | Entries in the same position combine with each other. |
Watch this section: review of matrix addition and subtraction at 2:57.
Timestamp Guide
Related Linear Algebra Work
For more matrix basics, browse the Matrices topic page. If you are moving from matrix arithmetic into row reduction, the Elementary Row Operations companion is a natural next step.
You can also use the JoeCMath Matrix/Linear Algebra playlist.