Question: Homework 4 Let A = () Question 1 Write out the quadratic characteristic polynomial f(1) of A explicitly. Question 2 Solve the characteristic polynomial f

Homework 4

Let A = () Question 1 Write out the quadratic characteristic polynomial f(1) of A explicitly. Question 2 Solve the characteristic polynomial f ()) for the two eigenvalues 1, and 12 of A. Question 3 Based on the eigenvalues of A in Question 2, is A invertible? Why or why not? Question 4 Find a unit eigenvector bj for the eigenspace Ex, and another unit eigenvector D2 for Ex2. Question 5 Let V be the 2 X 2 matrix with columns v1 and 12. Let D be a diagonal matrix with 1, and 12 on its diagonal. Verify A = VDV by matrix multiplication. Question 6 Verify tr(A) = 21 + 12. Question 7 Verify det(A) = 21 X 12. Question 8 What is the relationship between f(0) and det(A)? (This relationship is true in general.)

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