1)

{1 point) Suppose that A = [42 34] = LDU where: 1 G L = [ 1] is a lower triangular matrix with ones on the diagonal, a. [la I1] _ _ _ D = IS a diagonal matrix, and [l c 1 d U = [ 1] is an upper triangular matrix 1mm ones on the diagonal. Find L, D and U. (1 point) A matrix A is said to be similar to a matrix B if there is an invertible matrix P such that B = PAP-1. Let A1, A2, and A3 be 3 x 3 matrices. Prove that if A, is similar to A2 and Ag is similar to A3, then A, is similar to A3. Proof: Since A1 is similar to A2. for some invertible matrix P. Since A2 is similar to A3. for some invertible matrix Q Substituting the expression for A2 in the first equation into the second equation yields which is equivalent to for invertible matrix QP. This proves that Aj is similar to A3- Q. E. D. A. A1 = P(QA;Q 1)p-1 B. As = QA2Q 1 C. A2 = PAP-1 D. A2 = QA3Q 1 E. A1 = (QP)A3(QP)-1 F. A3 = (QP)A, (QP)-1 G. A3 = Q(PAP 1)Q 1 H. A1 = PA,P-1(1 point) An n x n matrix A is: i) symmetric if A" = A and li) skew-symmetric if -A? = A. Suppose that B is an n x n skew-symmetric matrix (i.e. -B* = B) and let I be the n x n identity matrix. Show that the product (I + B)(I - B) is symmetric. Proof: [(I + B)(I - B)]T Q. E. D. A. (I + B) (I - B) B. (I - B)T C. (IT - BT) D. (BT - IT) E. (I + B) T F. - (BT + IT) G. [(I + B) (I - B)IT H. (IT + BT)(1 point) The 2 x 2 elementary matrix E can be by found applying the row operation 4R2 + R1 -> R, to the identity matrix. Find EA if: -2 4 A = 1 5 EA =(1 point) Give a 4 x 4 elementary matrix E which will carry out the row operation 6R2 + Rs -> Rs. E = Test that E actually works for carrying out this row operation by computing the product EA for the matrix A = 4 2 EA =(1 point) -2 -2 -4 -4 (a) Suppose that E1 = . Then: 4 -3 4 -3 E1 = and Ejl = (b) Suppose that E2 Them: E2 = and E2 1 -2 -2 14 -14 (c) Suppose that E3 = . Then: 4 -3 4 -3 E3 = and Egl =