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Tool to calculate matrix additions in computer algebra. The sum of N matrices is generally obtained by summing the elements of each matrix.

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### What is a matrix addition? (Definition)

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: $$\forall i, j \quad c_{ij} = a_{ij}+b_{ij}$$

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

### How to add 2 matrices?

The addition of matrix is done entry by entry

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

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

### How to add 2 matrices in Excel?

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

### How to add 2 matrices of different sizes?

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.

### How to add a scalar to a matrix?

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.

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