Give a counterexample to show that $(AB)^{-1} \neq A^{-1}B^{-1}$

Question

Give a counterexample to show that $(AB)^{-1}$ doesn't equal $A^{-1}B^{-1}$

I'm not sure how to approach this, so I just used the idea that the matrix multiplication is not commutative. so it goes:

AB doesn't equal BA

now I just take the inverse of both sides if they are invertible (lets say they are)

so I get

$B^{-1}A^{-1}$ doesn't equal $A^{-1}B^{-1}$

meaning that the above is true.

I'm not good at doing proofs and as such my logic here is probably wrong so please someone verify if this is a method I could use to prove such.

Answer 1

You're almost there. Find any two invertible matrices $A$ and $B$ that do not commute i.e. $AB \neq BA$. Taking inverse on both sides, $(AB)^{-1}\neq (BA)^{-1}$

Answer 2

What you need now is an example of two invertible matrices that commute. How about $$ A = \pmatrix{1&0\\0&-1}, \quad B = \pmatrix{1&1\\0&1} $$