Really Quick question determinant properties

Question

I think there is a mistake in my textbook but just making sure

If $A$ and $B$ are $4 \times 4$ matrices and $\det(A) = 3$ and $\det(B) = 5$, what is the determinant of $(3A)^{-1}$?

My solution:

$$\frac{1}{\det(3A)} = \frac{1}{3^4 \det(A)} = \frac{1}{3^5}.$$ Their answer is $\frac{1}{3^4}.$

Answer 1

You are absolutely right.

There must be a mistake in the correction.

You always have

$$\det(CD)=\det(C)\det(D)$$

and

$$\det(A^{-1})=\frac 1{\det(A)}.$$

Answer 2

Your answer is correct. Maybe there is mistake in book.

$$det(3A)^{-1} = \frac{1}{\det(3A)}$$

And rest of the solution done by you perfectly.