Suppose that a 2×2 matrix A has an eigenvalue 2 with corresponding eigenvector [1 −1] and an eigenvalue −2 with corresponding eigenvector [3 −2]. Find an invertible matrix P and a diagonal matrix D so that A=PDP−1. Enter your answer as an equation of the form A=PDP−1. You must enter a number in every answer blank for the answer evaluator to work properly.
Question
Suppose that a 2×2 matrix A has an eigenvalue 2 with corresponding eigenvector [1 −1] and an eigenvalue −2 with corresponding eigenvector [3 −2]. Find an invertible matrix P and a diagonal matrix D so that A=PDP−1. Enter your answer as an equation of the form A=PDP−1. You must enter a number in every answer blank for the answer evaluator to work properly.
Solution
The matrix P is formed by placing the eigenvectors of A as columns. So, we have:
P = [[1, 3], [-1, -2]]
The matrix D is a diagonal matrix with the eigenvalues on the diagonal. So, we have:
D = [[2, 0], [0, -2]]
The inverse of P, denoted as P^-1, can be calculated as follows:
P^-1 = 1/(1*-2 - (-1*3)) * [[-2, -3], [1, 1]] = -1/(-2 + 3) * [[-2, -3], [1, 1]] = [[2, 3], [-1, -1]]
So, we have:
A = PDP^-1 = [[1, 3], [-1, -2]] * [[2, 0], [0, -2]] * [[2, 3], [-1, -1]]
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