We can do both the operations adding and subtracting matrices, if the order of the given matrices are equal i.e. order of A = order of B. If $A_{m\times n}$ = [aij] and $B_{m\times n}$ = [bij] are the given matrices, then matrix addition and subtraction are possible.
A = $\begin{bmatrix}
a_{11} &a_{12} \\
a_{21} &a_{22}
\end{bmatrix}$ and B = $\begin{bmatrix}
b_{11} &b_{12} \\
b_{21} &b_{22}
\end{bmatrix}$

$A\pm B = \begin{bmatrix}
a_{11}\pm b_{11} &a_{12}\pm b_{12} \\
a_{21}\pm b_{21} & a_{22}\pm b_{22}
\end{bmatrix}$ = $\begin{bmatrix}
c_{11} &c_{12} \\
c_{21} &c_{22}
\end{bmatrix}$ = C

Matrix addition is possible only if the order of both the matrices are equal. This is one of the matrix addition rules. To understand more about matrix addition, lets have a look at some examples.

Solved Examples

Question 1: If A = $\begin{bmatrix}
8 & 5 \\
4 & 3
\end{bmatrix}$ and B = $\begin{bmatrix}
7 & 0 \\
2 & -1
\end{bmatrix}$ then calculate A + B
Solution:
  1. Check the order of the matrices.
  2. Perform the addition of the matrices.

A + B = $\begin{bmatrix}
8 + 7 & 5 + 0 \\
4 + 2 & 3 + (-1)
\end{bmatrix}$

C = $\begin{bmatrix}
15 & 5 \\
6 & 2
\end{bmatrix}$

Question 2: If A = $\begin{bmatrix}
2 &3 &-7 \\
4 &9 &8 \\
1 &5 &-6
\end{bmatrix}$ , B = $\begin{bmatrix}
1 &3 \\
5 &7 \\
-6 &-1
\end{bmatrix}$ and C = $\begin{bmatrix}
-1 &3 &8 \\
5 &-9 &4 \\
0 &4 &-1
\end{bmatrix}$, calculate (i) A + C (ii) B + C.
Solution:
(i) Matrices A and C are of same order. So, matrix addition is possible.
A + C = $\begin{bmatrix}
2& 3 & -7 \\
4& 9 & 8\\
1& 5 &-6
\end{bmatrix}$ + $\begin{bmatrix}
-1 &3 &8 \\
5 &-9 &4 \\
0 &4 &-1
\end{bmatrix}$

= $\begin{bmatrix}
1& 6 & 1 \\
9 &0 &12 \\
1 &9 &-7
\end{bmatrix}$

(ii) We have matrices, B = $\begin{bmatrix}
1 &3 \\
5 &7 \\
-6&-1
\end{bmatrix}$ and C = $\begin{bmatrix}
-1 &3 &8 \\
5 &-9 &4 \\
0 &4 &-1
\end{bmatrix}$

Here, the order of B is $3\times 2$ and the order of C is $3\times 3$. For matrix addition, the order of the matrices must be the same. So, it is not possible to add B and C.

Question 3: Find the value of x and y in the relation $\begin{bmatrix}
3 &2x \\
y-1 &3
\end{bmatrix}$ + $\begin{bmatrix}
1 &x \\
4 & 6
\end{bmatrix}$ = $\begin{bmatrix}
4 &12 \\
6 &9
\end{bmatrix}$
Solution:
$\begin{bmatrix}
3 &2x \\
y-1 &3
\end{bmatrix}$ + $\begin{bmatrix}
1 &x \\
4 &6
\end{bmatrix}$ = $\begin{bmatrix}
4 &12 \\
6 &9
\end{bmatrix} $

By the use of addition property,
3 + 1 = 4, 2x + x = 12, y - 1+ 4 = 6, 3 + 6 = 9
So, we get 3x = 12 $\Rightarrow $ x = 4
and y + 3 = 6 $\Rightarrow $ y = 3

Matrix subtraction is possible only if the order of both the matrices are equal. This is one of the rules for subtracting matrices. To understand more about matrix subtraction, lets have a look at some examples.

Solved Examples

Question 1: Calculate A - B, if A = $\begin{bmatrix}
9& 0 & -5 \\
3& 9 &5\\
2& 2 &-1
\end{bmatrix}$ and B = $\begin{bmatrix}
7& 3 &6 \\
4& 9 & 6\\
1& 8 &-2
\end{bmatrix}$
Solution:
A - B = $\begin{bmatrix}
9& 0 & -5 \\
3& 9 &5\\
2& 2 &-1
\end{bmatrix}$ - $\begin{bmatrix}
7& 3 &6 \\
4& 9 & 6\\
1& 8 &-2
\end{bmatrix}$

= $\begin{bmatrix}
9-7 &0- 3 &-5-6 \\
3-4 & 9-9 & 5-6\\
2-1 & 2-8 &-1-(-2)
\end{bmatrix}$

= $\begin{bmatrix}
2 &- 3 &-11 \\
-1 & 0 &-16\\
1 &-6 &1
\end{bmatrix}$

Question 2: Calculate A - B, if A = $\begin{bmatrix}
5 & 3 & 0 &1 \\
7 & 4 & -5 &1 \\
8 & 2 & 9 &1 \\
6 & 1 & 8 & 1
\end{bmatrix}$ and B = $\begin{bmatrix}
2 & 7 & 1 &-1 \\
3 & 6 & 3 &2 \\
4 & 1 & 5 &0 \\
-7 & -8 & 5 & 0
\end{bmatrix}$
Solution:
A - B = $\begin{bmatrix}
5 - 2 & 3 - 7 & 0 - 1 &1 + 1 \\
7 - 3 & 4 - 6 & -5 - 3 &1 - 2 \\
8 - 4 & 2 - 1 & 9 - 5 &1 - 0 \\
6 + 7 &1 + 8 &8 - 5 & 1 - 0
\end{bmatrix}$
= $\begin{bmatrix}
3 & -4 & -1 &2 \\
4 & -2 & -8 &-1 \\
4 & 1 & 4 &1 \\
13 &9 &3 & 1
\end{bmatrix}$