Add and Subtract Matrices

In order to add and subtract matrices, they must have the same dimensions - or the same number of rows and columns. If two matrices have that, then you may add or subtract corresponding entries.

equation image indicator $LaTeX: A=\begin{bmatrix} 1 & -2 & 5\\ 0 & 4 & -1 \end{bmatrix} B= \begin{bmatrix} 2 & 0 & 4\\ 3 & -1 & 5 \end{bmatrix}$

Matrices A and B are both size 2 x 3 so I can add or subtract them. Let's try adding them:

equation image indicator $LaTeX: A+B=\begin{bmatrix} 1 & -2 & 5\\ 0 & 4 & -1 \end{bmatrix} + \begin{bmatrix} 2 & 0 & 4\\ 3 & -1 & 5 \end{bmatrix} = \begin{bmatrix} 1+2 & -2+0 & 5+4\\ 0+3 & 4+-1 & -1+5 \end{bmatrix} = \begin{bmatrix} 3 & -2 & 9\\ 3 & 3 & 4 \end{bmatrix}$

Let's try subtracting them:

equation image indicator $LaTeX: A-B=\begin{bmatrix} 1 & -2 & 5\\ 0 & 4 & -1 \end{bmatrix} - \begin{bmatrix} 2 & 0 & 4\\ 3 & -1 & 5 \end{bmatrix} = \begin{bmatrix} 1-2 & -2-0 & 5-4\\ 0-3 & 4--1 & -1-5 \end{bmatrix} = \begin{bmatrix} -1 & -2 & 1\\ -3 & 5 & -6 \end{bmatrix}$

Let's try a few:

equation image indicator $LaTeX: A=\begin{bmatrix} -4 & 7 & 9\\ 0 & 3 & 2 \end{bmatrix} B= \begin{bmatrix} 3 & 6 & 8\\ -1 & -2 & 0 \end{bmatrix} C=\begin{bmatrix} 1 & 0 & 5\\ 4 & 2 & -2\\ 7 & -3 & 1 \end{bmatrix} D= \begin{bmatrix} 7 & -2 & 0\\ 5 & 3 & 3\\ 6 & -1 & 4 \end{bmatrix}$

Notice up above that A + B = B + A. This means that matrix addition (like addition with real numbers) is commutative!

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