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.

 

[Architeuthis Dux]

There are a few ways to approach this problem, depending on the specific properties and definitions that have been covered in your class. Here are two possible explanations:

1. Using the distributive property of matrix multiplication:

We know that matrix multiplication is not commutative, meaning that AB is not always equal to BA. Therefore, we cannot simply cancel the B's in the equation AB^2 = (A^2)(B^2). However, we can use the distributive property to show that AB^2 is equal to A(BB), and (A^2)(B^2) is equal to (AA)(BB). Since matrix multiplication is associative, we can then show that A(BB) is equal to (AA)B, but not necessarily equal to A(AB). Therefore, the claim is not always valid.

2. Using counterexamples:

We can also provide a counterexample to show that the claim is not always true. Consider the matrices A = [1 0] and B = [0 1]. It is easy to verify that AB^2 = [0 1] but (A^2)(B^2) = [1 0], which are not equal. This counterexample shows that the claim is not always valid.

In general, it is important to understand the properties and definitions of matrix operations in order to prove or disprove claims like this one. It is also important to carefully consider any assumptions that are made, such as invertibility, in order to ensure the validity of the proof.