Nontrivial solution

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March 6th 2009, 03:26 PM
antman

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 $A^{-1}$ exists and multiplying both sides of Ax=0 by $A^{-1}$ gives $A^{-1}(Ax)=A^{-1}(0)$ so $(AA^{-1})(x)=0$ then $(I_n)x=0$ 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?
March 6th 2009, 05:44 PM
o_O

Quote:

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

And the second red is probably a typo.
March 7th 2009, 07:44 PM
antman

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