If $\mathrm{A}$ and $\mathrm{B}$ are square matrices of the same order such that $\mathrm{AB}=\mathrm{BA}$, prove that

$\left(\mathrm{A}+\mathrm{B}\right)\left(\mathrm{A}-\mathrm{B}\right)={\mathrm{A}}^2-{\mathrm{B}}^2$

If $\mathrm{A}$ and $\mathrm{B}$ are square matrices of the same order such that $\mathrm{AB}=\mathrm{BA}$, prove that

${\left(\mathrm{A}+\mathrm{B}\right)}^2={\mathrm{A}}^2+2\mathrm{AB}+{\mathrm{B}}^2$

If $\mathrm{A}$ and $\mathrm{B}$ are square matrices of the same order such that $\mathrm{AB}=\mathrm{BA}$, prove that

${\left(\mathrm{A}-\mathrm{B}\right)}^2={\mathrm{A}}^2-2\mathrm{AB}+{\mathrm{B}}^2$

If $\mathrm{A}$ and $\mathrm{B}$ are square matrices of the same order such that $\mathrm{AB}=\mathrm{BA}$, prove that

${\left(\mathrm{A}+\mathrm{B}\right)}^3={\mathrm{A}}^3+3{\mathrm{A}}^2\mathrm{B}+3{\mathrm{AB}}^2+{\mathrm{B}}^3$

If $\mathrm{A}$ and $\mathrm{B}$ are square matrices of the same order such that $\mathrm{AB}=\mathrm{BA}$, prove that

${\left(\mathrm{A}-\mathrm{B}\right)}^3={\mathrm{A}}^3-3{\mathrm{A}}^2\mathrm{B}+3{\mathrm{AB}}^2-{\mathrm{B}}^3$

Find $x$, if

$\left[\begin{array}{ccc}1& x& 1\end{array}\right]\left[\begin{array}{ccc}1& 3& 2\\ 2& 5& 1\\ 15& 3& 2\end{array}\right]\left[\begin{array}{c}1\\ 2\\ x\end{array}\right]=\mathrm{O}$

Find $x$, if 

$\left[\begin{array}{ccc}1& 2& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 0\\ 2& 0& 1\\ 1& 0& 2\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]=\mathrm{O}\left\{\right\}$.