Show that the matrices A= [ ll 1 & 2 3 & 1 ], B= [ cc 1 & -2 -3 &… - Vidyadip

Question

Show that the matrices $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}1 & -2 \\ -3 & 1\end{array}\right]$ satisfy commutative property $A B=B A$

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Answer

$
\begin{aligned}
& A=\left[\begin{array}{ll}
1 & 2 \\
3 & 1
\end{array}\right], B=\left[\begin{array}{cc}
1 & -2 \\
-3 & 1
\end{array}\right] \\
& A B=\left[\begin{array}{ll}
1 & 2 \\
3 & 1
\end{array}\right] \times\left[\begin{array}{cc}
1 & -2 \\
-3 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1-6 & -2+2 \\
3-3 & -6+1
\end{array}\right] \\
& =\left[\begin{array}{cc}
-5 & 0 \\
0 & -5
\end{array}\right] \ldots(1) \\
& BA =\left[\begin{array}{cc}
1 & -2 \\
-3 & 1
\end{array}\right] \times\left[\begin{array}{cc}
1 & 2 \\
3 & 1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1-6 & 2-2 \\
-3+3 & -6+1
\end{array}\right] \\
& =\left[\begin{array}{cc}
-5 & 0 \\
0 & -5
\end{array}\right] \ldots(2)
\end{aligned}
$

From (1) and (2) we get
$
A B=B A .
$
It satisfy the commutative property.

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