Given the matrices A and B below:
[tex]$A=\mleft[\begin{array}{cc}2 & 0 \\ -3 & 3\end{array}\mright]$,$B=\mleft[\begin{array}{cc}-1 & -3 \\ 0 & 2\end{array}\mright]$[/tex]
Part B
[tex]\begin{gathered} BA=\mleft[\begin{array}{cc}-1 & -3 \\ 0 & 2\end{array}\mright]\mleft[\begin{array}{cc}2 & 0 \\ -3 & 3\end{array}\mright] \\ $=\mleft[\begin{array}{cc}(-1\times2)+(-3\times-3) & (-1\times0)+(-3\times3) \\ (0\times2)+(2\times-3) & (0\times0)+(2\times3)\end{array}\mright]$ \\ $=\mleft[\begin{array}{cc}-2+9 & 0-9 \\ 0-6 & 0+6\end{array}\mright]$ \\ $BA=\lbrack\begin{array}{cc}7 & -9 \\ -6 & 6\end{array}\rbrack$ \end{gathered}[/tex]
Part C
[tex]\begin{gathered} $B=\mleft[\begin{array}{cc}-1 & -3 \\ 0 & 2\end{array}\mright]$ \\ \implies B^2=\mleft[\begin{array}{cc}-1 & -3 \\ 0 & 2\end{array}\mright]\mleft[\begin{array}{cc}-1 & -3 \\ 0 & 2\end{array}\mright] \\ =\mleft[\begin{array}{cc}(-1\times-1)+(-3\times0) & (-1\times-3)+(-3\times2) \\ (0\times-1)+(2\times0) & (0\times-3)+(2\times2)\end{array}\mright] \\ =\mleft[\begin{array}{cc}1+0 & 3-6 \\ 0+0 & 0+4\end{array}\mright] \\ \implies B^2=\mleft[\begin{array}{cc}1 & -3 \\ 0 & 4\end{array}\mright] \end{gathered}[/tex]
