4. Let A and B be two n × n matrices. Suppose that AB is invertible. Show that the system Ax = 0 has only the trivial solution
19. Suppose A and B are n xn matrices. a. Suppose that both A and B are diagonalizable and that they have the same eigen- vectors. Prove that AB = BA. b. Suppose A has n distinct eigenvalues and AB = BA. Prove that every eigen vector of A is also an eigen vector of B. Conclude that B is diagonalizable. (Query: Need every eigenvector of B be an eigenvector of A?)
4. Let A and B be two nx n matrices. Suppose that AB is invertible. Show that the system A.x = 0 has only the trivial solution.
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
1. Let A and B be two nx matrices. Suppose that AB is invertible. Show that the system Az = 0 has only the trivial solution. 5. Given that B and D are invertible matrices of orders n and p respectively, and A = W X1 Find A-" by writing A-as a suitably partitioned matrix B
. If A and B are n x n matrices such that the product AB is not invertible, then either A or B is not invertible. (We call such non-invertible matrices singular.)
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...
Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A
Linear algebra . For two matrices A and B, the product AB is an n × m1 m atrix and the product BA is a Show A and B must be squ
11. Prove one of the following: a. Let A and B be square matrices. If det(AB) + 0, explain why B is invertible. b. Suppose A is an nxn matrix and the equation Ax = 0 has a nontrivial solution. Explain why Rank A<n.