Chapter 3 - Determinants - 3.4 Summary of Determinants - Problems - Page 241: 21

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Given $A$ is an ivertible $n\times n$ matrix Then we get $A^{-1}A=I_n\\ \rightarrow \det(A^{-1}A)=\det(I_n)\\ \rightarrow \det(A^{-1})\det(A)=\det(I_n)\\ \rightarrow \det(A^{-1})\det(A)=1$ We know that $A$ is invertible so $\det(A)\ne 0$ Hence, $\det(A^{-1})=\frac{1}{\det(A)}$