Define L: R(mxm) to R(nxn). If L(A)=L(B), prove or disprove that det(A)=det(B).
The Attempt at a Solution
I think I can prove that this is true.
L(A)=L(B) means that L(A)-L(B)=L(A-B)=0.
Now let C be the matrix representation of L. We have two possibilities:
1) C is nonsingular. If C is nonsingular, then C(A-B)=0, so A-B=0. Then det(A)-det(B)=0 and det(A)=det(B).
2) C is singular. Now I have an issue-- I don't know what to do!
Am I on the right path? Should I be disproving this?