$A,B$ are two $6 \times 6$ real matrices and $A$ is invertible. I need to prove that if $\operatorname{adj} A$ is similar to $\operatorname{adj} B$ then $A$ is similar to $B$.
$\operatorname{adj}A = M^{-1}(\operatorname{adj}B) M$ but can't think of a direction to the solution?
Best Answer
Hint: use the relations \begin{align} A\operatorname{adj}A&=\det(A)I,\tag{1}\\ B\operatorname{adj}B&=\det(B)I\tag{2} \end{align} to prove first that $\det A=\det B$.
One way to do that: prove that $$ \det A\ne 0\implies \det\operatorname{adj}A\ne 0\implies \det\operatorname{adj}B\ne 0\implies \det B\ne 0. $$ Then take the determinant of boths side in (1) and (2).