Can you help solve this please

MAT22A Exercise 1. For each statement below prove it or give a counterexample. (a) If A is 3 x 3 upper triangular and B is 3 x 3 lower triangular, then the diagonal entries of AB are the products of the diagonal entries of A and B. (b) If A and B are both 3 x 3 upper triangular matrices, then the diagonal entries of AB are the products of the diagonal entries of A and B. (c) If D is an 3 x 3 diagonal matrix and A is an 3 x 3 matrix, then AD = DA. (d) If c is a scalar and A is an n x n matrix, then (c/) A = A(c/). [As usual, / denotes the identity matrix of the appropriate size.Exercise 2. Suppose that an LU-decomposition of a matrix A is 0 A = LU = 3 2 Use the two step method of forward-substitution and back-substitution from class and section 9.1 to solve AT = b where 4 6 = NO CT2 2 2 Exercise 3. Let A = 7 13 10 4 -22 (a) Compute an LU-decomposition of A: Find a lower triangular matrix L and an upper triangular matrix U such that A = LU. Remember LU-decompositions, even when they exist, might not be unique. (b) Find matrices , and Uj where L, is lower triangular with 1's down the diagonal and Uj is upper triangular such that A = LIU1. (c) Find matrices L2 and U2 where U2 is upper triangular with 1's down the diagonal and L2 is lower triangular such that A = L2U2. (d) Compute the LDU-decomposition of A: Find matrices Lo, Uo, and D where Lo is lower triangular with 1's down the diagonal, Vo is upper triangular with 1's down the diagonal, and D is a diagonal matrix such that A = LoDUo. Remember LDU-decompositions, when they exist, are unique