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!