True or false and proof Suppose that A is an m x n matrix su

True or false and proof

Suppose that A is an m x n matrix such that AX = B has a solution for all B R^n. Then the solution to A X = B^\' when it exists, is unique. Suppose that A is an m x n matrix such that the solution to when it exists, is unique. Then has a solution for all B R^m. Suppose that A is an m x n matrix such that AX = B has a solution for all Then the equation has a solution for all B\' R^n.

Solution

If A is an mxn matrix such that AX = B has a solution for all B R^m , then the solution to A^TX = B^T is unique.

If A is a mxn matrix, then A^T is a nxm matrix. Similarly if since B is a mx1 matrix, B^T is a 1xm matrix. Also, X is a nx1 matrix. If A^TX = B^T   has a solution, then n = m as the number of rows in X has to be equal to the number of columns of A^T . Further, since A^T X will be a m x 1 matrix, and B^T isa 1xm matrix, if A^TX = B^T   has a solution, then m = n = 1. Then A^TX = B^T is reduced to a single linear equation of the form ax = b in 1 variable, which will have a unique solution x = b/a when a 0.

If A is an mxn matrix such that the solution to A^TX = B^T is unique, then AX = B has a solution for all B R^m.

If A is an mxn matrix, then A^T is a nxm matrix. Therefore, if A^TX exists , then X will be a mx1 matrix so that A^TX will be a nx1 matrix. Then B^T will be a   nx1 matrix. However, B is a mx1 matrix as B R^m , so that B^T is a 1x m matrix. Therefore m = n = 1. Then AX = B will be a slinear equation of the form ax = b in 1 variable, which will have a unique solution x = b/a when a 0.

If A is an mxn matrix such that AX = B has a solution for all B R^m , then the equation A^TX = B^T has a solution for all B^T Rⁿ

If A is a mxn matrix, then A^T is a nxm matrix. Also, X is a nx1 matrix and B is a mx1 matrix as B R^m. Since A^T is a nxm matrix , for A^TX to exist m has to be equal to n Then A^TX is a nx1 matrix . Further, since A^TX = B^T and B^T Rⁿ , the matrix equation A^TX = B^T   represents a system of n linear equations in n variables which will have a solution for all B^T Rⁿ . The proposition will be FALSE also when B^T is a zero vector and the RREF of A is not a zero matrix.