Determinant of a $4\times4$ invertible matrix Let $A$ be a $4$ by $4$ invertible matrix, such that $\det(3A)=3\det(A^4)$.
Then $\det(A)=3$.
Would somebody please give me some clues on this?
Thanks

Recall the fact that $$\det(AB) = \det(A) \det(B)\tag{1}$$ and $$\det(k A) = k^n \det(A)\tag{2}$$ where $k \in \mathbb{C}$ and $A \in \mathbb{C}^{n \times n}$.
From $(1)$, we get that $\det(A^4) = \det(A)^4$ and using $(2)$, we get that $\det(3A) = 3^4 \det(A)$. Hence, we get that $$3^4 \det(A) = 3 \det(A)^4 \implies \det(A)^3 = 3^3 \implies \det(A) = 3,3 \omega \text{ or }3w^2.$$ Assuming $A$ has only real entries, the determinant also has to be real and hence $$\det(A) = 3$$
