Matrices Proof> C=A-B, if Ax=Bx where x is nonzero, show C is singular

[Luxe]

Homework Statement

Let A and B be n x n matrices and let C= A - B.
Show that if Ax=Bx, and x does not equal zero, then C must be singular.

Homework Equations

The Attempt at a Solution

Ax-Bx=0
x(A-B)=0
x(C)=0
So, Cx=0

Does that mean C is singular?

 

[Dick]

If Cx=0 and x is not the zero vector, then what would C^(-1)(0) be? C0=0 as well. Would it be x or 0? Sure, it means C is singular.