Determine the eigenvalues and eigenvector(s) for each matrix.

1) Determine the eigenvalues and eigenvector(s) for each matrix.

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1. a) (8 pts)
2. b) (9 pts)

2) (8 pts) The trace of a square matrix is the sum of its diagonal entries. This is written tr(A) = ∑a_ii. Square matrices have a neat property where the trace of the matrix is equal to the sum of its eigenvalues. Also, the determinant of a matrix is equal to the product of its eigenvalues. Written in math speak:

tr(A) = ∑λ_i

det(A) = ∏λ_i

Use the trace and the determinant properties to get a system of two (nonlinear) equations in λ₁ and λ₂ and solve for those eigenvalues for the matrix . HINTs: You’ll need to use the quadratic formula. Also, don’t be surprised if your eigenvalues are complex.

3) Give all of the eigenspaces for each matrix. Give the algebraic and geometric multiplicity associated with each eigenvalue.

1. b) (15 pts)

4) (5 pts) Compute PD²P^-1and A². What do you notice?

5) (5 pts) Compute . Is it equal to ?

6) (10 pts) Give a general equation to compute A^k. This works for every diagonalizable matrix A. Why does this work? Recall that A = PDP^-1.