Calculate determinant of the power of a matrix

Question

I am given a square matrix $A$ with determinant $-t$. My professor asks what’s the determinant of $\text{det}(A^{2020})$.

This is what I’ve done:

$$\text{det}(A^{2020})=(-1)^{2020}\text{det}(A)=-t$$

I’ve used $\text{det}(A^n)=(-1)^n \text{det}(A)$

This is what my professor worked out:

$$\text{det}(A^{2020})=\text{det}(A)^{2020}=(-t)^{2020}=t^{2020}$$

Are they both valid?

Answer 1

$\det(A^m)=(\det(A))^m$, The answer should be $t^{2020}$. Your Prof. is right.

Answer 2

You seem to believe that somehow $\;A^{2020}=(-1)^{2020}A\;$ , which is certainly false. What your prof. wrote is right. You are not multiplying by $\;-1\;$ the matrix $\;A\;$ ...!

Don't let that minus sign in $\;-t\;$ fool you. What you must realize is that is always true that $\;\det(A^k)=(\det A)^k\;$ ...that's all.