1. ## 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 $\displaystyle A^{-1}$exists and multiplying both sides of Ax=0 by $\displaystyle A^{-1}$ gives $\displaystyle A^{-1}(Ax)=A^{-1}(0)$ so $\displaystyle (AA^{-1})(x)=0$ then $\displaystyle (I_n)x=0$ 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?

2. 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 $\displaystyle A^{-1}$exists and multiplying both sides of Ax=0 by $\displaystyle A^{-1}$ gives $\displaystyle A^{-1}(Ax)=A^{-1}(0)$ so $\displaystyle ({\color{red}A^{-1}A})(x)=0$ then $\displaystyle (I_n)x=0$ 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 $\displaystyle A^{-1}$ from the left so you should stay consistent (even though $\displaystyle AA^{-1} = A^{-1}A = I$).

And the second red is probably a typo.

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