Is $\det(ABA)= \det(B)$?
Solution 1:
There are matrices $A$ such that $A=A^{-1}$. More generally if $\det A=\pm 1$ then $\det(ABA)=\det B$ for every $B$ (in fact the last statement is "if and only if")..
There are matrices $A$ such that $A=A^{-1}$. More generally if $\det A=\pm 1$ then $\det(ABA)=\det B$ for every $B$ (in fact the last statement is "if and only if")..