Finding the determinant of a matrix given it's $n \times n$ and that to the 20th power, it is singular.

Question

I have a question about homework problems that I'm going through with linear algebra at the moment.

I'm stuck on a question where it defines $A$ as an $n\times n$ matrix, where $A^{20}$ is singular.

How would I go about finding $\det(A)$?

Many thanks.

Answer 1

$A^{20}$ is singular means $\det(A^{20})=0$.

$\det$ is multiplicative, so $\det(A^{20})=\det(A)^{20}$.

Now do you know what $\det(A)=?$

Answer 2

Any matrix with a singular power must itself be singular. In particular, since $A^{20}$ is singular, so must $A$, which means that $\det(A)$ is zero.