Would someone please instruct me how to show if
$\displaystyle det(AB) = det(BA)$
Also, how should I show if rank(A)rank(B) is equal to rank(AB) or not.
$\displaystyle rank(AB)=rank(A)rank(B)$
Thank you.
$\displaystyle \det(AB)=\det(A)\det(B)=\det(B)\det(A)=\det(BA)$
It is false, choose for example $\displaystyle A=\begin{bmatrix}{0}&{1}\\{0}&{0}\end{bmatrix} \quad B=\begin{bmatrix}{0}&{2}\\{0}&{0}\end{bmatrix}$Also, how should I show if rank(A)rank(B) is equal to rank(AB) or not.