Help proving matrix properties:

[SrEstroncio]

Homework Statement

Let A, B be both matrices with the same dimensions. Is AB^2 = (A^2)(B^2) a valid claim?

Homework Equations

The Attempt at a Solution

I attempted to show that (AB)^2 = (AB)(AB) = A(BA)B
and that (A^2)(B^2) = (AA)(BB) = A(AB)B, so for A(BA)B to be equal to A(AB)B, AB must be equal to BA, which is not always true.

I discarded this approach as nothing assures me that A and B are both invertible, and thus I cannot prove that A(BA)B = A(AB)B implies BA = AB. My teacher is kinda picky about this stuff.

 

[Mark44]

In fact, it's not generally true that AB = BA, so if you can find a counterexample (start with 2x2 matrices), you will have shown that (AB)² = A²B² is not a valid claim.