Question

: Linear Algebra: If A and B are similar matrices, prove that det(A) = det(B).

Answer 1

If A and B are similar, then \[\Large B = P^{-1}AP\] where P is some invertible matrix


So


\[\Large \det(A)=\det(B)\]


\[\Large \det(A)=\det(P^{-1}AP)\]


\[\Large \det(A)=\det(P^{-1})\det(A)\det(P)\]


\[\Large \det(A)=\det(A)\det(P^{-1})\det(P)\]


\[\Large \det(A)=\det(A)\det(P^{-1}P)\]


\[\Large \det(A)=\det(A)\det(I)\]


\[\Large \det(A)=\det(A)(1)\]


\[\Large \det(A)=\det(A)\]


We can follow this backwards (since each statement is equivalent) to show that because \[\Large \det(A)=\det(A)\] is always true, this would mean that \[\Large \det(A)=\det(B)\] is always true when A and B are similar matrices

Answer 2

..oh wow I feel stupid now. HAHAHA. thank you very much!!