Matrix - Addition

Matrix addition is only well-defined for matrices of the same dimensions.

To add two matrices $A$ and $B$ with the same dimensions, just add the corresponding entries, e.g.,

$$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)+\left(\begin{array}{ccc}7& 8& 9\\ 10& 11& 12\end{array}\right)=\left(\begin{array}{ccc}1+7& 2+8& 3+9\\ 4+10& 5+11& 6+12\end{array}\right)=\left(\begin{array}{ccc}8& 10& 12\\ 14& 16& 18\end{array}\right)$$

Or more formally using indices, if $A$ and $B$ are both $m$ by $n$ matrices, then their sum $C=A+B$ is also an $m$ by $n$ matrix defined by

$$C_{i,j}=A_{i,j}+B_{i,j}\text{for }1\le i\le m,1\le j\le n$$