Thread: Nontrivial solution

How do I prove / disprove "If a Matrix A is nonsingular, then the homogeneous system Ax=0 has a nontrivial solution?"

I know that if A is nonsingular, then exists and multiplying both sides of Ax=0 by gives so then and x=0. Therefore, the only solution to Ax=0 is x=0 so a nonsingular matrix never has a trivial solution to Ax=0.

Is this enough to disprove the statement?

Originally Posted by antman

How do I prove / disprove "If a Matrix A is nonsingular, then the homogeneous system Ax=0 has a nontrivial solution?"

I know that if A is nonsingular, then exists and multiplying both sides of Ax=0 by gives so then and x=0. Therefore, the only solution to Ax=0 is x=0 so a nonsingular matrix never has a nontrivial solution to Ax=0.

Is this enough to disprove the statement?

Just a minor change. One, you're multiplying both sides by from the left so you should stay consistent (even though ).

And the second red is probably a typo.

Thank you! That does make more sense and that definitely was a typo.