When we want to add and subtract matrices together, each matrix has to be the same size.
So they need to have the same number of rows and the same number of columns.
A 3 × 3 matrix would have to be added to or subtracted from another 3 × 3 matrix.
We just add together the equivalent matrix elements.
\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}
= \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \\ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \\ a_{31}+b_{31} & a_{32}+b_{32} & a_{33}+b_{33} \end{bmatrix}
The same goes for matrix subtraction, it’s the same approach when we add and subtract matrices.
Add and Subtract Matrices Examples
(1.1)
\begin{bmatrix} 1 & 2 & 4 \\ 3 & 1 & 8 \end{bmatrix} + \begin{bmatrix} 5 & 6 & 7 \\ 4 & 1 & 1 \end{bmatrix}
= \begin{bmatrix} 1+5 & 2+6 & 4+7 \\ 3+4 & 1+1 & 8+1 \end{bmatrix} = \begin{bmatrix} 6 & 8 & 11 \\ 7 & 2 & 9 \end{bmatrix}
(1.2)
\begin{bmatrix} 3 & 2 & 1 \\ 4 & 4 & 2 \\ 9 & 8 & 6 \end{bmatrix} − \begin{bmatrix} 1 & 1 & 2 \\ 2 & 3 & 1 \\ 7 & 5 & 4 \end{bmatrix}
= \begin{bmatrix} 3-1 & 2-1 & 1-2 \\ 4-2 & 4-3 & 2-1 \\ 9-7 & 8-5 & 6-4 \end{bmatrix} = \begin{bmatrix} 2 & 1 & {\text{-}}1 \\ 2 & 1 & 1 \\ 2 & 3 & 2 \end{bmatrix}
(1.3)
\begin{bmatrix} {\text{-}}1 & 3 \\ 4 & {\text{-}}9 \\ 2 & 7 \end{bmatrix} + \begin{bmatrix} {\text{-}}2 & 1 \\ 3 & 0 \\ 1 & 2 \end{bmatrix}
= \begin{bmatrix} {\text{-}}1+{\text{-}}2 & 3+1 \\ 4+3 & 3+0 \\ 2+1 & 7+2 \end{bmatrix} = \begin{bmatrix} {\text{-}}3 & 4 \\ 7 & 3 \\ 3 & 9 \end{bmatrix}
(1.4)
\begin{bmatrix} {\text{-}}2 & 5 & 4 \\ 0 & {\text{-}}2 & 6 \end{bmatrix} − \begin{bmatrix} {\text{-}}4 & 3 & 1 \\ {\text{-}}1 & {\text{-}}7 & 9 \end{bmatrix}
= \begin{bmatrix} {\text{-}}2-{\text{-}}4 & 5-3 & 4-1 \\ 0-{\text{-}}1 & {\text{-}}2-{\text{-}}7 & 6-9 \end{bmatrix}
= \begin{bmatrix} 2 & 2 & 3 \\ 1 & 5 & {\text{-}}3 \end{bmatrix}
Multiplying a Matrix by a Number
When multiplying a matrix by a single number, all we need to do is multiply each matrix element by that number.
The matrix itself stays the same size with the same number of rows and columns.
Examples
(2.1)
3 × \begin{bmatrix} 4 & 1 & 2 \\ 3 & 5 & 1 \end{bmatrix} = \begin{bmatrix} 3\times4 & 3\times1 & 3\times2 \\ 3\times3 & 3\times5 & 3\times1 \end{bmatrix} = \begin{bmatrix} 12 & 3 & 6 \\ 9 & 15 & 3 \end{bmatrix}
(2.2)
–2 × \begin{bmatrix} {\text{-}}1 & 4 \\ 7 & {\text{-}}3 \\ 0 & 8 \end{bmatrix} = \begin{bmatrix} {\text{-}}2\times1 & {\text{-}}2\times4 \\ {\text{-}}2\times7 & {\text{-}}2\times{\text{-}}3 \\ {\text{-}}2\times0 & {\text{-}}2\times8 \end{bmatrix}
= \begin{bmatrix} {\text{-}}2 & {\text{-}}8 \\ {\text{-}}14 & 6 \\ 0 & {\text{-}}16 \end{bmatrix}