Matrix Addition

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

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

The addition of matrix is done entry by entry

For all matrices A and B of the same size, A+B = B+A.

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 (all dimensions possible, provided they are exactly the same: 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.