[Math] Finding possible determinant values of 3×3 matrix using an equation

Question

Given a 3 x 3 matrix $A$

$4A= A^{7}$

Find the possible values of det(A). I multiplied by $A^{-1}$ both sides and got
$4I= A^{6}$ (not helpful) ??
Can you show the right way to solve it ? and how does knowing that it's 3×3 help us ?

Answer 1

If $4A=A^7$, then $\det(4A)=\det(A^7)=[\det(A)]^7=4^3\det(A)$

Since $\det(cA)=c^n\det(A)$

Let $\det(A)=x$, then we must solve $64x=x^7$