Question

Show that if A is both diagonalizable and invertible, then so is A __________.

A. By mathematical proof
B. Through experimental data
C. Based on theoretical assumptions
D. Using graphical analysis

Answer 1

Option B. If a matrix A is diagonalizable and invertible, its inverse A^-1 is also diagonalizable. we can find a diagonal matrix D' and an invertible matrix P' such that A^-1 = P'D'P'^-1.

Step-by-step explanation:

If a matrix A is diagonalizable and invertible, then it can be expressed as a product of a diagonal matrix D and an invertible matrix P, such that A = PDP^-1. Since A is invertible, its inverse exists, denoted by A^-1. To show that A^-1 is also diagonalizable, we can find a diagonal matrix D' and an invertible matrix P' such that A^-1 = P'D'P'^-1.

Let's start by finding the inverse of A. We have A^-1 = (PDP^-1)^-1. Using the property that (AB)^-1 = B^-1A^-1, we can rewrite this as A^-1 = (P^-1)^-1D^-1P^-1. Since P^-1 is invertible, (P^-1)^-1 exists, and we can simplify further as A^-1 = PDP^-1.

Now, let's examine the expression A^-1 = PDP^-1. It has the same form as the expression A = PDP^-1, which shows that A^-1 is also diagonalizable, with the same diagonal matrix D.