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Linear AlgebraProof

The determinant of an inverse

Let $A$ be an invertible $n \times n$ matrix. Prove that $\det A \neq 0$ and

\det(A^{-1}) = \frac{1}{\det A}.

You may use that $\det(AB) = \det(A)\det(B)$ for all $n \times n$ matrices $A$ and $B$ .

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