Real or Complex Matrix Addition, Subtraction, and Scalar Multiplication

Similar to vectors, we can add or subtract matrices. This is done by adding or subtracting the corresponding entries of each matrix.  

Note that this can only be done for matrices of the same dimensions.  

Consider $A = \begin{bmatrix} 1 & -2 \\ 5 + i & 3 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & i \\ 7 & -2 \end{bmatrix}.$

Then:

$$\begin{align*} A + B &= \begin{bmatrix} 1 & -2 \\ 5 + i & 3 \end{bmatrix} + \begin{bmatrix} -1 & i \\ 7 & -2 \end{bmatrix} \\ &= \begin{bmatrix} 1 - 1 & -2 + i \\ (5 + i) + 7 & 3 - 2 \end{bmatrix} \\ &= \begin{bmatrix} 0 & -2 + i \\ 12 + i & 1 \end{bmatrix}. \end{align*}$$

Similarly, we can calculate $A - B$:

$$\begin{align*} A - B &= \begin{bmatrix} 1 & -2 \\ 5 + i & 3 \end{bmatrix} - \begin{bmatrix} -1 & i \\ 7 & -2 \end{bmatrix} \\ &= \begin{bmatrix} 1 - (-1) & -2 - i \\ (5 + i) - 7 & 3 - (-2) \end{bmatrix} \\ &= \begin{bmatrix} 2 & -2 - i \\ -2 + i & 5 \end{bmatrix}. \\ \end{align*}$$

In index notation, we would express this as:

\begin{align*} ...