Rank and Determinant of A*B matrix

**itpro** · December 30th 2011, 07:38 PM

Would someone please instruct me how to show if

$det(AB) = det(BA)$

Also, how should I show if rank(A)rank(B) is equal to rank(AB) or not.
$rank(AB)=rank(A)rank(B)$

Thank you.

**FernandoRevilla** · December 30th 2011, 10:59 PM

Originally Posted by itpro

Would someone please instruct me how to show if $det(AB) = det(BA)$

$\det(AB)=\det(A)\det(B)=\det(B)\det(A)=\det(BA)$

Also, how should I show if rank(A)rank(B) is equal to rank(AB) or not.

It is false, choose for example $A=\begin{bmatrix}{0}&{1}\\{0}&{0}\end{bmatrix} \quad B=\begin{bmatrix}{0}&{2}\\{0}&{0}\end{bmatrix}$

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