Matrix Addition, Subtraction, and Multiplication by a Scalar
MATRIX ADDITION AND SUBTRACTION
First of all, to add or subtract matrices, they must be the same size as one another!! This is very important! (You can't add or subtract if the sizes are different.) Actually doing the addition or subtraction is quite easy...you just add/subtract the corresponding entries in each matrix.
Example: If \(
A= \begin{bmatrix}
5 & 1 & 7 \\
2 & -1 & 3 \\
\end{bmatrix}
\) and \( B = \begin{bmatrix}
3 & 4 & 2 \\
6 & -8 & -5 \\
\end{bmatrix}
\)
Solution:
\[
\begin{equation}
\begin{split}
A + B & = \begin{bmatrix}
5+3 & 1+4 & 7+2 \\
2+6 & -1-8 & 3-5 \\
\end{bmatrix} \\
& = \begin{bmatrix}
8 & 5 & 9 \\
8 & -9 & -2 \\
\end{bmatrix} \\
\end{split}
\end{equation}
\]
\begin{equation}
\begin{split}
A + B & = \begin{bmatrix}
5+3 & 1+4 & 7+2 \\
2+6 & -1-8 & 3-5 \\
\end{bmatrix} \\
& = \begin{bmatrix}
8 & 5 & 9 \\
8 & -9 & -2 \\
\end{bmatrix} \\
\end{split}
\end{equation}
\]
MULTIPLICATION BY A SCALAR
To multiply a matrix by some scalar (i.e. number) \(c\), you simply multiply each entry in the matrix by \(c\).
Example: If
\(
A = \begin{bmatrix}
5 & 1 & 7 \\
2 & -1 & 3
\end{bmatrix}
\), find 3A
Solution:
\[\begin{equation}
\begin{split}
3A & = \begin{bmatrix}
(3)(5) & (3)(1) & (3)(7) \\
(3)(2) & (3)(-1) & (3)(3) \\
\end{bmatrix} \\
& = \begin{bmatrix}
15 & 3 & 21 \\
6 & -3 & 9 \\
\end{bmatrix} \\
\end{split}
\end{equation}
\]
Example:
