Question

Would this statement be true regarding matrices: For ANY m x n matrices A and B, A+B = B+A, or would this only be true if the matrices were of the same size?

Answer 1

If `A` and `B` have the same dimensions `m xx n`, then matrix addition is indeed commutative.

Explanation:

Matrix addition `A+B` is only defined when `A` and `B` are conformal. For addition, that means they must have the same number of rows and columns as each other.

Matrix addition is performed by adding corresponding elements of `A` and `B` to get the new elements of `A+B`. If `A` and `B` have different sizes, then there will be elements of one that don't have a partner in the other. In this case, the sum `A+B` does not exist.

If `A` and `B` have the same dimensions, then the addition `A+B` produces the same sum as `B+A`, because classic addition is also commutative. (That means `a+b=b+a`.)