Is there a general rule of thumb for calculating determinants?

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For example if we're told matrices A and B are both 3x3 matrices with det(A)=2 and det(B)=4.
I know that det(AB) = 8

Also det(3$A^2$)=108, but I don't understand how or why that is.

But for example what would be det($B^{-1}$ $A^T$).

So what im really wondering is if theres a general formula or a way to get the answers.

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Hint :

$$det(AB)=det(A)det(B)$$

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

$$det(A^T)=det(A)$$

If $A$ is a $n\times n$-matrix, we have $$det(mA)=m^ndet(A)$$