Matrix Addition

The addition of matrices can only be done with 2 matrices of the same shape (size).

If \( M_1=[a_{ij}] \) is a matrix of \( m \) lines and \( n \) columns (with \( m \) and \( n \) identical in the case of a square matrix) and \( M_2=[b_{ij}] \) a matrix also of \( m \) lines and \( n \) columns.

The sum of these 2 matrices \( M_1 + M_2 = [c_{ij}] \) is a matric of the same size, ie. \( m \) lines and \( n \) columns, with : $$ \forall i, j : c_{ij} = a_{ij}+b_{ij} $$

Example: $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} + \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} = \begin{bmatrix} 1+7 & 2+8 \\ 3+9 & 4+10 \\ 5+11 & 6+12 \end{bmatrix} = \begin{bmatrix} 8 & 10 \\ 12 & 14 \\ 16 & 18 \end{bmatrix} $$

Add the elements to the identical coordinates in each matrix.

The addition operation (or sum) for matrices can only be done with same size matrices. Nevertheless, there is the direct sum operation, which can be used with distinct size matrices.