# Basic Linear Algebra Proof

1. Jul 18, 2009

1. The problem statement, all variables and given/known data

Prove the following:

For every n × n complex matrices A and B, $$(\alpha AB)^*=\bar{\alpha }B^*A^*$$.

2. Relevant equations

None

3. The attempt at a solution

Okay, I'm just getting started on this problem. All the ideas I have come up with so far involve using two "test" matrices. The problem with this is that it doesn't prove it for any n × n matrix. Does this matter?

2. Jul 18, 2009

### Office_Shredder

Staff Emeritus
Ok, start small. Can you prove

$$(\alpha A)^*=\bar{\alpha }A^*$$

If you can, then you can just focus on proving that A and B swap like that

3. Jul 18, 2009

Hmm...

This feels like trying to prove 1+1=2. It just is! I'm still working on it...

4. Jul 18, 2009

$$(\alpha AB)^*=\bar{\alpha }\overline{AB}=\bar{\alpha }\left(\bar{A}\right)\left(\bar{B}\right)=\bar{\alpha }\left(\left(\bar{A}\right)^T\right)^T\left(\left(\bar{B}\right)^T\right)^T=\bar{\alpha }\left(A^*\right)^T\left(B^*\right)^T=\bar{\alpha }\left(B^*A^*\right)^T$$