If A= [ lll 5 & 2 & 9 1 & 2 & 8 ], B= [ cc 1 & 7 1 & 2 5 & -1 ] verify that (A B)^ =B^ A^

& Given A= [ lll 5 & 2 & 9 1 & 2 & 8 ], B= [ cc 1 & 7 1 & 2 5 & -1 ] & A B= [ lll 5 & 2 & 9 1 & 2 & 8 ] [ cc 1 & 7 1 & 2 5 & -1 ] & = [ cc 5+2+45 & 35+4-9 1+2+40 & 7+4-8 ] & = [ cc 52 & 30 43 & 3 ] & ( AB )^ = [ cc 52 & 43 30 & 3 ] (1) & B ^ = [ ccc 1 & 1 & 5 7 & 2 & -1 ] & A ^ = [ cc 5 & 1 2 & 2 9 & 8 ] & B ^ A ^ = [ ccc 1 & 1 & 5 7 & 2 & -1 ] [ ll 5 & 1 2 & 2 9 & 8 ] & = [ cc 5+2+45 & 1+2+40 35+4-9 & 7+4-8 ] & = [ cc 52 & 43 30 & 3 ] (2) From (1) and (2) we get, (A B)^ =B^ A^

If A= [ lll 5 & 2 & 9 1 & 2 & 8 ], B= [ cc 1 & 7 1 & 2 5 & -1 ] v…

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Question

If $A=\left[\begin{array}{lll}5 & 2 & 9 \\ 1 & 2 & 8\end{array}\right], B=\left[\begin{array}{cc}1 & 7 \\ 1 & 2 \\ 5 & -1\end{array}\right]$ verify that $(A B)^{\top}=B^{\top} A^{\top}$

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Answer

$\begin{aligned} & \text { Given } A=\left[\begin{array}{lll}5 & 2 & 9 \\ 1 & 2 & 8\end{array}\right], B=\left[\begin{array}{cc}1 & 7 \\ 1 & 2 \\ 5 & -1\end{array}\right] \\ & A B=\left[\begin{array}{lll}5 & 2 & 9 \\ 1 & 2 & 8\end{array}\right] \times\left[\begin{array}{cc}1 & 7 \\ 1 & 2 \\ 5 & -1\end{array}\right] \\ & =\left[\begin{array}{cc}5+2+45 & 35+4-9 \\ 1+2+40 & 7+4-8\end{array}\right] \\ & =\left[\begin{array}{cc}52 & 30 \\ 43 & 3\end{array}\right] \\ & ( AB )^{\top}=\left[\begin{array}{cc}52 & 43 \\ 30 & 3\end{array}\right] \ldots(1) \\ & B ^{\top}=\left[\begin{array}{ccc}1 & 1 & 5 \\ 7 & 2 & -1\end{array}\right] \\ & A ^{\top}=\left[\begin{array}{cc}5 & 1 \\ 2 & 2 \\ 9 & 8\end{array}\right]\end{aligned}$
$
\begin{aligned}
& B ^{\top} A ^{\top}=\left[\begin{array}{ccc}
1 & 1 & 5 \\
7 & 2 & -1
\end{array}\right] \times\left[\begin{array}{ll}
5 & 1 \\
2 & 2 \\
9 & 8
\end{array}\right] \\
& =\left[\begin{array}{cc}
5+2+45 & 1+2+40 \\
35+4-9 & 7+4-8
\end{array}\right] \\
& =\left[\begin{array}{cc}
52 & 43 \\
30 & 3
\end{array}\right] \ldots(2)
\end{aligned}
$
From (1) and (2) we get, $(A B)^{\top}=B^{\top} A^{\top}$