Currently workable: Let A be an n x n invertible matrix. Suppose AB n x p....
Currently workable: Suppose and m x n matrix A has n pivot columns. Prove why, for each b in R the equation Ax = b has at most one solution. + Drag and drop your files or click to browse...
Suppose A is a square matrix such that det A4 invertible. 0. Prove that A is not Suppose that A is a square matrix such that det A" invertible and that it must have determinant 1. 1. Prove that A is Matrices whose determinant is 1 are part of a group (not just the english word, a special math term, ask if you want the deets) called the Special Linear Group, denoted SL(n) + Drag and drop your files or...
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)...
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = p-1AP is the diagonal form of A. Prove that A* = Pokp-1, where k is a positive integer. Use the result above to find the indicated power of A. 10 18 A = -6 -11 18].46 A = 11
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = P-1AP is the diagonal form of A. Prove that Ak = Pekp-1, where k is a positive integer. Use the result above to find the indicated power of A. 0-2 02-2 3 0 -3 ,45 A5 = 11
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = p-1AP is the diagonal form of A. Prove that Ak = pokp-1, where k is a positive integer. Use the result above to find the indicated power of A. -10 -18 A = 6 11 18].45 -253 -378 A6 = 126 188 11
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 be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = −4 0 4 −3 −1 4 −6 0 6 , A5
5. Find a 2 x 2 matrix A such that A2 = I2, but A + +12. (Hint: you can do this algebraically, or geometrically.) For all the remaining questions, let n > 2 and let A and B be n x n matrices. 6. Does the equation A(B – In) + (In – B)A = On,n always hold? Either prove it or give a counter-example. 7. If A and B are invertible, does that imply that AB is invertible?...
Suppose A and B are matrices with matrix product AB. If bi, b2, ..., br are the columns of B, then Ab, Ab2, ..., Ab, are the columns of AB 1. Suppose A is an nxnmatrix such that A -SDS where D diag(di,d2,... dn) is a diagonal matrix, and S is an invertible matrix. Prove that the columns of S are eigenvectors of A with corresponding eigenvalues being the diagonal entries of D Before proving this, work through the following...