Answer
Because of theorem 6 and theorem 4
Work Step by Step
Theorem 6 states that for all square matrices, the following holds:
$$\det(AB)=\det(A)\det(B)$$
Since $A$ is a square matrix and:
$$A^3=A\cdot A \cdot A$$
We have:
$$\det(A^3)=\det(A)\det(A)\det(A)=0$$
The only number cubed that equals $0$ is $0$, therefore it holds that $\det(A)=0$, which, by theorem 4, makes it not invertible.
