<s>[INST] Are following statements about matrices true? 1) If $AB+BA=0$, then $A^2B^3=B^3A^2$
2) If $A$ and $B$ are non-singular, then $AB$ is non-singular
3) If $A^3=0$, then $A-I$ is non-singular
$A$ and $B$ are $n \times n$ matrices and $I$ is identity matrix.
And for 3), If square of matrix is $0$, then its matrix has to be zero element? 
 [/INST] For 1, $AB+BA=0\implies AB=-BA$. Using this, we can try and turn the LHS $A^2 B^3$ into $B^3A^2$ by succcessively replacing $AB$ with $-BA$.
\begin{align*}
A^2B^3 &= A(AB)B^2 \\
&= A(-BA)B^2 \\
&= -(AB)(AB)B \\
&= -(-BA)(-BA)B \\
&= -B(AB)(AB) \\
&= -B(-BA)(-BA) \\
&= -B^2(AB)A \\
&= -B^2(-BA)A \\
&= B^3A^2\:.
\end{align*}
For 2, a matrix is nonsingular iff its determinant is nonzero. If we know that $\det A,\det B\neq0$, this implies that
$$\det(AB)=(\det A)(\det B)\neq0.$$
For 3, we take inspiration from the factorisation of $x^3-1=(x-1)(x^2+x+1)$. Since $A^3=0$, this means that $I-A^3=I$. It can be verified by expansion that
$$(A-I)(-A^2-A-I)=(-A^2-A-I)(A-I)=I-A^3=I$$
which implies that $A-I$ is invertible, and has inverse $-A^2-A-I$.
</s>