Show that if A is both diagonalizable and invertible, then so is A-1. What does

Question

Show that if A is both diagonalizable and invertible, then so is A-1. What does it mean if A is diagonalizable? If A is diagonalizable, then A = PD for some invertible P and diagonal D. If a is diagonalizable, then Ak = PDP^-1 for some invertible P and diagonal D. if a is diagonalizable, then A = PDP^- 1 for some invertible P and diagonal D. If A is diagonalizable, then A must be a triangular matrix. What does it mean if A is invertible? A has no less than three eigenvalues, so the diagonal entries in D are not zero, so D is invertible. Zero is not an eigenvalue of A, so the diagonal entries in D are not zero, so D is invertible. Zero must be an eigenvalue of A, so at least one of the diagonal entries in D is zero, so D is invertible. O D- A has no more than three eigenvalues, so the diagonal entries in D are not zero, so D is invertible. What is the inverse of A? A^-1 = P^-1D^-1p A^-1 = PDP^-1 A^-1 = P^-1D^-1 A^-1 = PD^-1P^-1 Therefore, A^-1 is also diagonalizable.

Explanation / Answer

1) C

2) A

3) D

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