Chapter 5 - Eigenvalues and Eigenvectors - 5.3 Exercises - Page 289: 27

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A is diagonalizable, so there exists invertible matrix P and diagonal matrix D such that $A=PDP^{-1}$ A is invertible, so all its eigenvalues are nonzero. This means all the columns of D are nonzero and linearly independent, which means D is invertible. Thus, $A^{-1}=(PDP^{-1})^{-1}=PD^{-1}P^{-1}$