1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix...

1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix computation to verify that χA(A) = 0.

(4 3

-1 1) (2 1 -1 0 3 0 0 -1 2) (3*3 matrix)

2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ.

A= (9 42 -30 -4 -25 20 -4 -28 23),λ = 1,3
A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2

3. Construct a 2×2 matrix with eigenvectors(4 3) and (−3 −2) with eigen-values 2 and −3, respectively.

4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of λ as a root of the characteristic polynomial and compare it to the dimension of the eigenspace Eλ.

(x 0 0 0 0 0 0 x 0 0 0 0 0 0 y 0 0 0 0 0 0 x 0 0 0 0 0 0 z 0 0 0 0 0 0 x)

5. Let A be an 3*3 upper triangular matrix with all diagonal elements equal, such as (3 4 -2 0 3 12 0 0 3)

Prove that A is diagonalizable if and only if A is a scalar times the identity matrix.

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Expert Solution

Colby Messinger answered 2 years ago

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