true or false and proof Suppose that A is an mn matrix such

true or false and proof

Suppose that A is an m×n matrix such that AX=B has a solution for all BE R\". Then the solution to A\'X B\', when it exists, is unique. exists, is unique. Then AX = B has a solution for all B E R\'\" B e R\". Then the equation A\'X- B\' has a solution for all B\' E R\". e2 Suppose that A is an m × n matrix such that the solution to AX = B, when it 3 Suppose that A is an m×n matrix such that AX=B has a solution for all

Solution

1. When A is a mxn matrix, and X is a nx1 matrix, AX is a mx 1 matrix and so is B which R^m. Let us now look at the equation A^TX = B^T . Now, since A is a mxn matrix, A^T is a nxm matrix. Since X is a nx1 matrix, we must have n = m if the matrix A^TX has to exist. Then A^TX is a nx1 matrix. Since B is a mx1 matrix, B^T is a 1xm matrix. Since A^TX = B^T , the dimensions of A^TX has to be same as that of B^T. Therefore, m = n = 1. Then, the equation A^TX = B^T is a single linear equation of the form ax = b which has a unique solution x = - b/a if a 0

2. When A is a mxn matrix, A^T is a nxm matrix. For A^TX to exist, X has to be a mx1 matrix. Then A^TX is a nx1 matrix. However, since B R^{m ,} B is a mx1 matrix and therefore, B^T is a 1xm matrix.  Further, since A^TX = B^T , the matrices A^TX and B^T should have the same dimension. Therefore m = n =1. Then the equation AX = B is a single linear equation of the form ax = b which has a solution x = - b/a for all b R ( here R^m is R as m = 1) if a 0.

3. If AX = B has a solution for all B R^m . The matrix A^T is a nxm matrxix and X is a nx1 matrix. Thus for the matrix A^T X to exist, we must have n =m . Thus the matrix equation AX = B is equivalent to a system of n linear equations in n variables which will have a solution if the RREF of A is not a zero matrix and B is not a zero matrix. If so, A^TX = B^T will be equivalent to a system of n linear equations in n variables which will have a solution if the RREF of A^T is not a zero matrix and B^T is not a zero matrix. With these conditions, theproposition is true.