Matric addition is a arithmetic operation that is performed on two matrices. In more intuitive geometrical terms, matrix addition is adding image vector of two different maps.

There is only one rule of matrix addition. The matrices should have same dimensions. Because when adding matrices, we are adding corresponding elements of the two matrices. If the matrices have different sizes, the correspooding element won't exist in the other matrix to perform addition

For example, if there are two matrix. A and B

Matrix A has dimensions 2x3 (which is 2 rows and 3 columns) and B has dimensions 3x3 (3 rows and 3 columns)

Then, the matrix addition is not possible because A don't have 3rd row to add with 3rd row of B

To Perform matrix addition. Add corresponding elements of the two matrices. For example, if there are two matrices C and D

C = [1 2 3] and D = [4 5 6]

The new matrix E = C + D

= [1+4 2+5 3+6] = [5 7 9]

So, the new matrix E is [5 7 9]

The formula for matrix addition is

E = A + B

Where E is the new matrix, A and B are the matrices to be added

Matrix addition with different dimension is not possible. Because, matrix addition is simply addition of corresponding elements in matrix. So addition cannot exists if the corresponding won't exist

If the corresponding element is 0, addition is possible. Because zero is also a number where arithmetic operations (except for division) can be performed.

Matrix addition is commutative. That means, the order of matrices doesn't matter when adding two matrices.

For example, if there are two matrices A and B

A = [1 2 3] and B = [4 5 6]

Then, A + B = B + A

So [4 5 6] + [7 8 9] = [4 5 6] + [7 8 9] = [11 13 15]

Even if you switch the order and add (B + A), the result will be same as A + B

Associative property of matrix addition is also true. That means, the grouping of matrices doesn't matter when adding three matrices.

For example, if there are three matrices A, B and C

A = [1 2 3], B = [4 5 6] and C = [7 8 9]

Then, A + (B + C) = (A + B) + C

It does not matter in which order you group the matrices, the result will be same in all cases