A= [ ll 1 & 0 2 & 1 ] and B= [ cc 2 & 3 -1 & 0 ] Find A^2+A B+B^2

Question

$A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & 3 \\ -1 & 0\end{array}\right]$ Find $A^2+A B+B^2$

✓

Answer

Given that
$
\begin{aligned}
& A=\left[\begin{array}{ll}
1 & 0 \\
2 & 1
\end{array}\right] \\
& B=\left[\begin{array}{cc}
2 & 3 \\
-1 & 0
\end{array}\right] \\
& A^2=A \times A=\left[\begin{array}{ll}
1 & 0 \\
2 & 1
\end{array}\right] \times\left[\begin{array}{ll}
1 & 0 \\
2 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 \times 1+0 \times 2 & 1 \times 0+0 \times 1 \\
2 \times 1+1 \times 2 & 2 \times 0+1 \times 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1+0 & 0+0 \\
2+2 & 0+1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 & 0 \\
4 & 1
\end{array}\right] \\
& A \times B=\left[\begin{array}{ll}
1 & 0 \\
2 & 1
\end{array}\right] \times\left[\begin{array}{cc}
2 & 3 \\
-1 & 0
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 \times 2+0 \times-1 & 1 \times 3+0 \times 0 \\
2 \times 2+1 \times 1 & 2 \times 3+1 \times 0
\end{array}\right] \\
& =\left[\begin{array}{cc}
2 & 3 \\
3 & 6
\end{array}\right] \\
& B^2=B \times B=\left[\begin{array}{cc}
2 & 3 \\
-1 & 0
\end{array}\right] \times\left[\begin{array}{cc}
2 & 3 \\
-1 & 0
\end{array}\right] \\
& =\left[\begin{array}{cc}
2 \times 2+3 \times(-1) & 2 \times 3+3 \times 0 \\
-1 \times 2+0 \times(-1) & -1 \times 3+0 \times 0
\end{array}\right] \\
& =\left[\begin{array}{cc}
4-3 & 6+0 \\
-2+0 & -3+0
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 & 6 \\
-2 & -3
\end{array}\right] \\
& A^2+A B+B^2=\left[\begin{array}{cc}
1 & 0 \\
4 & 1
\end{array}\right]+\left[\begin{array}{cc}
2 & 3 \\
3 & 6
\end{array}\right]+\left[\begin{array}{cc}
1 & 6 \\
-2 & -3
\end{array}\right] \\
& =\left[\begin{array}{cc}
1+2+1 & 0+3+6 \\
4+3-2 & 1+6+-3
\end{array}\right] \\
& =\left[\begin{array}{cc}
4 & 9 \\
5 & 4
\end{array}\right]
\end{aligned}
$