Let A= [ lll 1 & 3 & 2 ]_ 1 3 and B= [ l 3 2 1 ]_ 3 1 , find A B. Does BA exist? If yes, find it.

Product AB is defined and order of AB is 1 . A B= [ lll 1 & 3 & 2 ] [ l 3 2 1 ] =[1 3+3 2+2 1] =[11]_ 1 1 Again since number of column of B= number of rows of A=1 The product BA also is defined and order of BA is 3 . BA= [ l 3 2 1 ]_ 3 1 [ lll 1 & 3 & 2 ]_ 1 3 3 = [ lll 3 1 & 3 3 & 3 2 2 1 & 2 3 & 2 2 1 1 & 1 3 & 1 2 ]_ 3 3 = [ ccc 3 & 9 & 6 2 & 6 & 4 1 & 3 & 2 ]_ 3 3

Let A= [ lll 1 & 3 & 2 ]_ 1 3 and B= [ l 3 2 1 ]_ 3 1 , find A B.…

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Let $A=\left[\begin{array}{lll}1 & 3 & 2\end{array}\right]_{1 \times 3}$ and $B=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]_{3 \times 1}$, find $A B$. Does $BA$ exist? If yes, find it.

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Answer

Product $AB$ is defined and order of $\mathrm{AB}$ is $1 .$
$\therefore A B=\left[\begin{array}{lll}1 & 3 & 2\end{array}\right]\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]$
$=[1 \times 3+3 \times 2+2 \times 1]$
$=[11]_{1 \times 1}$
Again since number of column of $B=$ number of rows of $\mathrm{A}=1$
$\therefore$ The product $BA$ also is defined and order of $BA$ is $3 .$
$\mathrm{BA}=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]_{3 \times 1}\left[\begin{array}{lll}1 & 3 & 2\end{array}\right]_{1 \times 3 \times 3}$
$=\left[\begin{array}{lll}3 \times 1 & 3 \times 3 & 3 \times 2 \\ 2 \times 1 & 2 \times 3 & 2 \times 2 \\ 1 \times 1 & 1 \times 3 & 1 \times 2\end{array}\right]_{3 \times 3}$
$=\left[\begin{array}{ccc}3 & 9 & 6 \\ 2 & 6 & 4 \\ 1 & 3 & 2\end{array}\right]_{3 \times 3}$

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