Can I multiply 2 determinants of square matrices when they are of different in sizes?

Question

In the image attached, you can see the problem. (I am supposed to compute iii) I know that any equally sized square matrices' determinants can be multiplied directly.$$det(AB) = det(A)det(B)$$ Here however one of the matrices is 4x4 and the other is 5x5 so I am not sure if that rule holds or if this value can be computed.

Answer 1

Regardless in what order you would try to multiply out $A^5 B^2$, the usual matrix product will not be defined. You might be dealing with some kind of exotic product there.

Answer 2

You can compute $\det(A^5)\det(B^2)= \det(A)^5\det(B)^2,$ but you can't compute $\det(A^5B^2)$ since you can't even multiply $A^5$ and $B^2$! (The number of columns of $A$ should be equal to the number of rows of $B$ to multiply them).