I am having a hard time with proving some statements that deal with matrix operations.

Here's what I'm working on.

1) A and B are both n by n matrices.

Show that the formula (A + B)^2 = A^2 + 2AB + B^2 is not valid in general....

Here's what I worked up so far....

On the LHS you have AA + AB + BA + BB when you foil out (A + B)^2

On the RHS you have AA + AB +AB + BB when you express the squares without exponents.

It simply isn't good enough to check a single matrix and show that

AA + AB + BA + BB is not equal to AA + AB + AB + BB

but I do know that AB and BA will have different results.

Secondly to prove the distributive property of matrices :

(c + d)A = cA +dA

Let A be an m x n matrix and let c and d be scalars

not really sure what to do here at all....

Lastly to prove the statement (A + B)C = AC + BC

Let A = m x n matrix, B = m x n matrix and C = n x p matrix

I'm even more unsure of how to prove this.... I would think that

A and B are of the same dimensions such that for example A and B are 3x2

matrices and C would be a 2x4 matrix... your resulting matrix would be 3x4 ?

Even still that has nothing to do with the actual proof ?

Anyways I would REALLY appreciate any help!