Question
If $A=\left[\begin{array}{ll}3 & 2 \\ 0 & 5\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 0 \\ 1 & 2\end{array}\right]$ find the each of the following and state it they are equal: $(A+B)(A-B)$
✓
Answer
Given
$
A=\left[\begin{array}{ll}
3 & 2 \\
0 & 5
\end{array}\right]
$
and
$
\begin{aligned}
& B=\left[\begin{array}{ll}
1 & 0 \\
1 & 2
\end{array}\right] \\
& (A+B)(A-B) \\
& =\left\{\left[\begin{array}{ll}
3 & 2 \\
0 & 5
\end{array}\right]+\left[\begin{array}{ll}
1 & 0 \\
1 & 2
\end{array}\right]\right\} \times\left\{\left[\begin{array}{ll}
3 & 2 \\
0 & 5
\end{array}\right]-\left[\begin{array}{ll}
1 & 0 \\
1 & 2
\end{array}\right]\right\} \\
& =\left[\begin{array}{ll}
3+1 & 2+0 \\
0+1 & 5+2
\end{array}\right] \times\left[\begin{array}{ll}
3-1 & 2-0 \\
0-1 & 5-2
\end{array}\right] \\
& =\left[\begin{array}{ll}
4 & 2 \\
1 & 7
\end{array}\right] \times\left[\begin{array}{cc}
2 & 2 \\
-1 & 3
\end{array}\right] \\
& =\left[\begin{array}{cc}
8-2 & 8+6 \\
2-7 & 2+21
\end{array}\right] \\
& =\left[\begin{array}{cc}
6 & 14 \\
-5 & 23
\end{array}\right] . \\
&
\end{aligned}
$
$
A=\left[\begin{array}{ll}
3 & 2 \\
0 & 5
\end{array}\right]
$
and
$
\begin{aligned}
& B=\left[\begin{array}{ll}
1 & 0 \\
1 & 2
\end{array}\right] \\
& (A+B)(A-B) \\
& =\left\{\left[\begin{array}{ll}
3 & 2 \\
0 & 5
\end{array}\right]+\left[\begin{array}{ll}
1 & 0 \\
1 & 2
\end{array}\right]\right\} \times\left\{\left[\begin{array}{ll}
3 & 2 \\
0 & 5
\end{array}\right]-\left[\begin{array}{ll}
1 & 0 \\
1 & 2
\end{array}\right]\right\} \\
& =\left[\begin{array}{ll}
3+1 & 2+0 \\
0+1 & 5+2
\end{array}\right] \times\left[\begin{array}{ll}
3-1 & 2-0 \\
0-1 & 5-2
\end{array}\right] \\
& =\left[\begin{array}{ll}
4 & 2 \\
1 & 7
\end{array}\right] \times\left[\begin{array}{cc}
2 & 2 \\
-1 & 3
\end{array}\right] \\
& =\left[\begin{array}{cc}
8-2 & 8+6 \\
2-7 & 2+21
\end{array}\right] \\
& =\left[\begin{array}{cc}
6 & 14 \\
-5 & 23
\end{array}\right] . \\
&
\end{aligned}
$
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