Let A, B, and C be invertible matricies. Solve the following matrix equation for X: ABCX=S
Let A, B, C and D be fixed n x n invertible matrices. Does the equation C(A - 2X)B =D have a solution for a n x n matrix X? If so, find it.
6. Suppose that A is a 3 3 diagonalizable matrix, so there is an invertible 3 x 3 matrix S and scalars a, b, c so that Let C1,C2, č3 be the columns of S. Use the equation S-1S - I3 to computeSč1, S-C2, S-1c3 Show that B[c1,c2,cs is an eigenbasis for A I3 to computeS-1a 6. Suppose that A is a 3 3 diagonalizable matrix, so there is an invertible 3 x 3 matrix S and scalars a, b,...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
Question 5 (a) Let S be a k × k invertible symmetric matrix, and C be a k × k invertible matrix. Moreover, let i be a k-dimensional vector. Show the following equality (b) Set 3 0 0 Calculate (Ca) (CSCT)-(Ca) and S. Does your answer contradict the claim in part (a)? Ex- plain Са?) and S-12. Does your answer contradict the claim in part (a)? Ex
2. Let A be an invertible n x n matrix, and let (v) E C be an eigenvector of A with corresponding eigenvalue X E C. (a) Show that +0. (b) Further show that v) is also an eigenvector of A- with corresponding eigenvalue 1/1.
For the following problems use: Annx n matrix A is invertible RREF(A) = I rank(A) - n A 2 x 2 matrix A is invertible = det(A) 0 3 singular (non-invertible). For which value(s) of h is A = -2 -1 -4 Choose... Choose... 6 2 h-2 a 0,b 0,c+0,d +0 A = 4 -1 C 0 x-2 or x 4 For which values of x is A = invertible a 0,b 0,c 0,d=0 4 x 2 X#1 and x2...
(9pts) If A, B,C are n x n matrices, solve for the n x n matrix X (a) AXB = C if A invertible (b) A-XTA= B if A is invertible (c) XB A +3.XB if B is invertible (9pts) If A, B,C are n x n matrices, solve for the n x n matrix X (a) AXB = C if A invertible (b) A-XTA= B if A is invertible (c) XB A +3.XB if B is invertible
Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB are linearly independent. Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB are linearly independent.
Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant. Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant.
(a) Show that if the matrix B is invertible, then the only solution of the equation BX = 0 (where 0 is the zero square matrix of the same size as B) is X-0. (b) Consider a matrix partitioned in blocks, of the form (Α (в ο). с) where A and C are invertible, not necessarily of the same size. Find its inverse, itself partitioned in blocks of the same size, in terms of A, B, C. Hint: one of...