Addition of matrices:

Two matrices can be added only when they are of the same order

If two matrices of A and B are of the same order, they are said to be conformable for addition.

If A and B are matrices of order m × n, then their sum will also be a matrix of the same order and in addition, corresponding elements of A and B get added.

So if $A={\left[a_{ij}\right]}_{m\times n},B={\left[b_{ij}\right]}_{m\times n}$ Then, $A+B={[a_{ij}+b_{ij}]}_{m\times n}$ for all $i,j$ eg

$\begin{array}{rl}A& =\left[\begin{array}{lll}10& 20& 30\\ 20& 30& 40\\ 30& 40& 50\end{array}\right],B=\left[\begin{array}{lll}50& 40& 30\\ 40& 30& 20\\ 30& 20& 10\end{array}\right]\\ A+B& =\left[\begin{array}{lll}10+50& 20+40& 30+30\\ 20+40& 30+30& 40+20\\ 30+30& 40+20& 50+10\end{array}\right]=\left[\begin{array}{lll}60& 60& 60\\ 60& 60& 60\\ 60& 60& 60\end{array}\right]\end{array}$

Subtraction of matrices:

Two matrices can be subtracted only when they are of the same order. If A and B are matrices of order m × n then their difference will also be a matrix of the same order and in subtraction, corresponding elements of A and B get subtracted. So if 

$A={\left[a_{ij}\right]}_{m\times n},B={\left[b_{ij}\right]}_{m\times n}$ Then, $A-B={[a_{ij}-b_{ij}]}_{m\times n}$ for all $i,j$ eg

$\begin{array}{rl}\mathrm{A}& =\left[\begin{array}{lll}10& 20& 30\\ 20& 30& 40\\ 30& 40& 50\end{array}\right],\mathrm{B}=\left[\begin{array}{lll}50& 40& 30\\ 40& 30& 20\\ 30& 20& 10\end{array}\right]\\ \mathrm{A}-\mathrm{B}& =\left[\begin{array}{lll}10-50& 20-40& 30-30\\ 20-40& 30-30& 40-20\\ 30-30& 40-20& 50-10\end{array}\right]=\left[\begin{array}{ccc}-40& -20& 0\\ -20& 0& 20\\ 0& 20& 40\end{array}\right]\end{array}$
 

Properties of matrix addition:

1. Matrix addition is commutative, $\mathrm{A}+\mathrm{B}=\mathrm{B}+\mathrm{A}$
2. Matrix addition is associative, $A+(B+C)=(A+B)+C$
3. Additive identity exist, means there exist a matrix $O$ (null matrix) such that $A+O=A=O+A$ (Here $O$ has same order as A )
4. Existence of additive inverse means there exists a matrix $B$ such that $A+B=O=B+A$
5. Cancellation property:

If $A+B=A+C$ then $B=C$
If $\mathrm{A}+\mathrm{C}=\mathrm{B}+\mathrm{C}$ then $\mathrm{A}=\mathrm{B}$
Note: all matrix taken in above property explanation has same order which is $\mathrm{m}\times \mathrm{n}$.

Scalar multiplication:

Let k be any scalar number, and $A={\left[a_{ij}\right]}_{m\times n}$  be a matrix. Then the matrix is obtained by multiplying every element A by a scalar k and denoted as kA.

$\begin{array}{rl}& kA={\left[k{a}_{ij}\right]}_{m\times n}\\ & \mathrm{A}=\left[\begin{array}{ll}2& 6\\ 3& 7\\ 5& 8\end{array}\right]\text{ then, }3\mathrm{A}=\left[\begin{array}{ll}3\times 2& 3\times 6\\ 3\times 3& 3\times 7\\ 3\times 5& 3\times 8\end{array}\right]=\left[\begin{array}{cc}6& 18\\ 9& 21\\ 15& 24\end{array}\right]\end{array}$

Properties of scalar multiplication: 

If $A$ and $B$ are two matrices and $k,1$ are scalar then
i) $k(A+B)=kA+kB$
ii) $kl(A)=k(IA)=l(kA)$
iii) $(k+1)A=kA+IA$
iv) $(-k)A=-(kA)=k(-A)$
v) $1\mathrm{A}=\mathrm{A},(-1)\mathrm{A}=-\mathrm{A}$

Note: $A$ and $B$ have the same order $m\times n$.