Matrix addition and subtraction
When we add or subtract two matrices A and B, we add or subtract the corresponding elements. So we can only add or subtract matrices that have the same dimensions. In general, the sum and difference of two matrices can be represented similar to the following: \begin{bmatrix} a&b\\ c&d \end{bmatrix} + \begin{bmatrix} e&f\\ g&h \end{bmatrix} = \begin{bmatrix} a+e&b+f\\ c+g&d+h \end{bmatrix} and\begin{bmatrix} a&b\\ c&d \end{bmatrix} - \begin{bmatrix} e&f\\ g&h \end{bmatrix} = \begin{bmatrix} a-e&b-f\\ c-g&d-h \end{bmatrix}\\
As an example, the sum and difference of two matrices with numerical elements will look like the following: \begin{bmatrix} 3&7\\ 0&4 \end{bmatrix} + \begin{bmatrix} 1&2\\ 9&6 \end{bmatrix} = \begin{bmatrix} 4&9\\ 9&10 \end{bmatrix} and\begin{bmatrix} 2&-7\\ 3&0 \end{bmatrix} - \begin{bmatrix} -1&6\\ 0&4 \end{bmatrix} = \begin{bmatrix} 3&-13\\ 3&-4 \end{bmatrix}\\
Examples
Example 1
Consider the matrices: A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \\ 4 & 5 \end{bmatrix} and B = \begin{bmatrix} 3 & 4 & 5 \\ 2 & 1 & 9 \end{bmatrix}
Matrix A has dimensions ⬚ \times ⬚
Matrix B has dimensions ⬚ \times ⬚
Is A+B possible?
Example 2
If A = \begin{bmatrix} 7 \\ -7 \\ -5 \end{bmatrix} and B = \begin{bmatrix} 8 \\ 2 \\ 4 \end{bmatrix}, find A+B.
We can add or subtract two matrices of the same dimensions by adding or subtracting the corresponding elements.