1. Determine if the following matrices are diagonalizable. If so, compute their eigenvalue decomposition. (a) A = 11-1 02-1 0 0 1 (b) A= 3 -2 8 -7 -3 3 -16 then det(AAI) = (A-1)³.

In Exercises 23-26, use the method of Example 4.5 to find all of the eigenvalues of the matrix A. Give bases for each of the corresponding eigenspaces. 25.

a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.

For what values of a does the matrix A=[01a1] have the characteristics below? a A has eigenvalue of multiplicity 2. b A has 1 and 2 as eigenvalues. c A has real eigenvalues.