QUESTION 6 a) Prove the product of 2 2 x 2 symmetric matrices A and B...
Question 18: If Aand Bare symmetric n xn matrices prove that AB is symmetric or give an example that ABneed not be symmetric.
6. Show that if A is an n x n symmetric matrix and B is an n x m matrix. Show that BT AB, BT B and BBT are symmetric matrices (10 pts)
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A).
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix,
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
3. -4 points 3.2.036. 0/3 Submlssions Used an example to show that if A and 8 are symmetric n x n matrices, then AB need not be symmetric. 1 2 1 0 0 1 0 2 0 1 OA-12 0 1 o] 1 0 0 1 (b) Prove that if A and B are symmetnic n n matrices, then AB is symmetric if and only if AB BA. Suppose A and B are symmetric. Then (AB)T ATs-AB. Thus, AB is...
Determine if the statements are true or false. 1. If A and B are nxn matrices and if A is invertible, then ABA-1 = B. ? A 2. If A and B are real symmetric matrices of size nxn, then (AB)? = BA 3. If A is row equivalent to B, then the systems Ax = 0 and Bx = 0 have the same solution. ? A 4. If, for some matrix A and some vectors x and b we...
A square matrix is called skew-symmetric if AT = -A. (a) (4 points) Explain why the main diagonal of a skew-symmetric matrix consists entirely of zeros. (b) (2 points) Provide examples of a 2 x 2 skew-symmetric matrix and a 3 x 3 skew-symmetric matrix. (6 points) Prove that if A and B are both n x n skew-symmetric matrices and c is a nonzero scalar, then A + B and cA are both skew-symmetric as well. (4 points) Find...
2. Partitioned matrices A matrix A is a (2 x 2) block matrix if it is represented in the form [ A 1 A2 1 A = | A3 A4 where each of the A; are matrices. Note that the matrix A need not be a square matrix; for instance, A might be (7 x 12) with Aj being (3 x 5), A2 being (3 x 7), A3 being (4 x 5), and A4 being (4 x 7). We can...
4. (5 points) Let A and B ben x n matrices. Prove that if A and B are skew symmetric, then A - B is skew symmetric. Recall C = [cj] is skew symmetric iff Cij =-Cji.
Help on Questions 1-3
Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...