–HW 10 –gilbert –1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page –find all choices before answering. 001 10.0 points If the matrix is diagonalizable, i.e., A = PDP−1 with P invertible and D diagonal, which of the following is a choice for D? 1. 2. A is not diagonalizable correct 3.4.5.Explanation:Since ,the eigenvalues of A are the solutions of 9 −6λ + λ2 = (3 −λ)2 i.e., λ = 3, 1.FALSE 2.TRUE correctExplanation:Consider the 3 × 3 triangular matrix Because A is triangular, its eigenvalues are the entries along the diagonal, i.e., λ = 5, −2. Since these are distinct, diagonalizable. On the other hand, one of its eigenvalues is zero, so A is not invertible (or note that det[A] = 0 because det[A] = 5(0)(−2) = 0 is the product of the diagonal values of A, so A is not invertible). Therefore, an n matrix A can be diagonalizable, but not invertible. Consequently, the statement is .TRUE
3. On the other hand, when λ = 3, rref(A−λI) = rref,so x2 is the only free variable. Thus the eigenspace Nul(A−3I) has dimension 1. But then, when λ = 3, geo multA(λ) < alg multA(λ).Consequently, .002 10.0 points An n×n matrix can be diagonalizable, but not invertible. True or False?
003 10.0 points If the matrix A is not diagonalizable