Is L(A) = P^-1AP an Invertible Linear Operator?

[hkus10]

Let P belongs to Mnn be a nonsingular matrix and Let L:Mnn>>>Mnn be given by L(A) = P^-1AP for all A in Mnn. Prove that L is an invertible linear operator.
I have no clues how to start this question.
What do I need to prove for this question? and why

 

[LCKurtz]

Generally, a function is invertible if it is 1-1. Show that.

 

[HallsofIvy]

I think the best way to show something is invertible is to show its inverse!
What is $(P^{-1}AP)(P^{-1}A^{-1}P)$? What is $(P^{-1}A^{-1}P)(P^{-1}AP)$.

Or, since these are matrices, you could show that its determinant is not 0.
$$det(P^{-1}AP}= det(P^{-1})det(A)det(P)$$

Or, because of that, any product of invertible matrices is invertible.