Let Q be an invertible n x n matrix A and n x n matrix and x

Let Q be an invertible n x n matrix, A and n x n matrix and x is an n x 1 column vector(or matrix) so that the matrix equation AX=0 represents a homogeneous system of n equations in n unknowns. Show that (QA)x=0 has just the trivial solution, then A is invertible.

Solution

A homogeneous system of linear equations Ax = 0 has only the trivial solution if and only if the matrix A is invertible. Similarly, the homogeneous system (QA)x = 0 has just the trivial solution if and only if the matrix QA is invertible. Therefore, if QA is invertible, then both Q and A are invertible.

Thus (QA)x=0 has just the trivial solution, then A is invertible.