Nontrivial Solutions in Homogeneous Linear Systems: A Proof Problem

[paulrb]

Homework Statement

Suppose that AX = O is a homogeneous system of n equations in n variables. If the system (A^2)X = O has a nontrivial solution, show that AX = O has a nontrivial solution.

Homework Equations

Reduced row echelon form definition, matrix multiplication, etc.

The Attempt at a Solution

This looks like it would be easier to prove the contrapositive:
If AX = O does not have a nontrivial solution, then (A^2)X does not have a nontrivial solution.

However I'm not sure how to solve this.
If AX = O does not have a nontrivial solution, then the bottom row of A in reduced row echelon form is not all 0's. Should I use that to prove the bottom of A^2 in reduced row echelon form is not all 0's? Because I'm having trouble with that. Or maybe there is a different way to prove this problem.

 

[Mark44]

From the given information, the matrix A is n x n. You're also given that A²x = 0 has a nontrivial solution, which has implications about the value of the determinant |A²|. Is that enough to get you started?