Alright im having some issues here with proofs (i think). I have the following problem.
Let A and B be nxn matrices and let C = A-B. Show that if A(Xo) = B(Xo) and (Xo) !=0 then C must be singular
I have attempted to prove it and am unsure if i have come to the correct conclusion, or if im even mathematically allowed to do what i have done
C = A - B
A(Xo) = B(Xo)
A(Xo)-B(Xo) = 0
C(Xo) = (A-B)Xo
C(Xo) = 0
Since Xo cannot be zero than C must be singular. Does that make sense???
In addition to that problem I have another one that im not sure if the proof is correct. The problem states....
Let A and B be nxn matrices and let C=AB. Prove that is B is singular then C must also be singular.
Attempt at a proof....
C(C^-1) = I
AB(C^-1) = I
C^-1 = I(AB)^-1
C^-1 = B^-1 * A^-1
Since we know that B is singular B^-1 is undetermined, meaning that C^-1 must also be undetermined. And that proves that C must be singular???? I think?
I feel like neither answer is correct....