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.
Best Answer
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, $(AB)^{-1}\neq (BA)^{-1}$