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!
Proving that an identity isn't valid in general means exactly that; there may be specific cases for which it is valid. You do only need to find one matrix for which the identity doesn't hold. You can't prove that matrix multiplication doesn't commute in every case because it's not true. and , where is the identity matrix of any order, commute.
As for associativity, there is another way. You can show that matrix multiplication corresponds (pretty much by definition) to composition of linear transformations, which are just special functions, and that function composition is associative. It's a more elegant way that avoids messy computations but it still takes some work.
In order to prove that a general statement is not true, you only have to show a single "counter example".At any rate thank you both very much for your responses!
Since, "commuting matrices" are, at any rate, rare, you might try just picking matrices at random. For example, suppose
(which I did, literally, choose at random)
What is A+ B? What is ? Is it the same as ? If not, you are done.