will rate lifesaver please help find the characteristic polynomial eigenvalues,e

Question

will rate lifesaver please help find the characteristic polynomial eigenvalues,eigenvectors and (if possible) an invertible matrix P such that P^-1AP IS DIAGONAL A=[1-23][26-6][12-1]

Explanation / Answer

First, find its eigenvalues. Solving |A - ?I| = 0: |1-?...1...1| |1....-?...-1| = 0 |1....0...1-?| Expand on last row: 1(-1+?) + (1-?)[(1-?)(-?) - 1] = 0 ==> (? - 1) [1 - (?^2 - ? - 1)] = 0 ==> (? - 1) (?^2 - ? - 2) = 0 ==> ? = 1, -1, 2. ---------------------- Now we find the corresponding eigenvectors. (i) ? = -1. We row reduce A - (-1)I = [2 1 1] [1 1 -1] [1 0 2] to [1 0 2] [0 1 -3] [0 0 0]; this has eigenvector u = (-2, 3, 1)^t. (ii) ? = 1. We row reduce A - (1)I = [0 1 1] [1 -1 -1] [1 0 0] to [1 0 0] [0 1 1] [0 0 0] which yields eigenvector v = (0, -1, 1)^t. (iii) ? = 2. We row reduce A - 2I = [-1 1 1] [1 -2 -1] [1 0 -1] to [1 0 -1] [0 1 0] [0 0 0], which has eigenvector w = (1, 0, 1)^t. Letting D be the diagonal matrix with entries -1, 1, 2 down the diagonal, we set P = [u, v, w] (as its columns).

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