Matrix Addition

The addition of matrices (matrix A plus matrix B) can only be done with 2 matrices of the same shape/size (2x2, 2x3, 3x2, 3x3, etc.).

The addition of 2 matrices is noted $ M_1 + M_2 $ with $ M_1=[a_{ij}] $ ($ m $ lines and $ n $ columns, with $ m = n $ for a square matrix) and $ M_2=[b_{ij}] $ (also of $ m $ lines and $ n $ columns).

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

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

A matrix addition in Excel can be achieved by adding the elements with identical coordinates in each matrix.

The addition operation (or sum) for matrices can only be done with same size matrices (3x4, 4x3, 4x4, 5x5, etc.). Nevertheless, there is the direct sum operation, which can be used with distinct size matrices.

The operation of adding a scalar number to a matrix $ [A] + b $ is not defined, but sometimes it implies the operation $ [A] + [I] b $ with $ I $ the identity matrix of size compatible with A.