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

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If Ax = Bx for a nonzero vector x, then it follows that Ax - Bx = 0, leading to x(A - B) = 0, which can be rewritten as Cx = 0 where C = A - B. Since x is nonzero and Cx = 0, this indicates that the matrix C must be singular. The discussion confirms that the existence of a nonzero solution to Cx = 0 implies C does not have an inverse. Thus, the conclusion is that C is indeed singular. This proof demonstrates the relationship between the matrices A, B, and C in terms of their singularity.
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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?
 
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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.
 

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