## Linear Algebra: A Modern Introduction

We know that if $A$ and $B$ are $n\times n$ matrices, then $det(AB)=det(A)det(B)$. We also know that if $A$ is invertible, then $det(A^{-1})=\frac{1}{det(A)}$ We also know that if $A$ is an $n\times n$ matrix, then $det(kA)=k^ndet(A)$ We also know that if $A$ is a square matrix, then $det(A^T)=det(A)$ Hence $det(B^{-1}AB)=det(B^{-1})det(A)det(B)=\frac{1}{det(B)}det(A)det(B)=det(A)$