If A - - 0 1 2- 1 2 3- 2 3 4 - and B - - 1 -2- -1 0- 2 -1 - Check AB-BA-

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Matrix Definition and Representation

If A = [ 0 ...

Question

If $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 2 1 & 2 & 3 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦$ and $B = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 2 - 1 & 0 2 & - 1 \end{matrix} ⎤ ⎥ ⎦$ , Check $A B = B A$ ?

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Solution

Here the number of columns in $A = 3$ = the number of rows in $B$ .
Therefore the product AB is defined.
$A B = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 2 1 & 2 & 3 2 & 3 & 4 \end{matrix} ⎤ ⎥ ⎦ \begin{matrix} R_{1} R_{2} R_{3} \end{matrix} \times ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} C_{1} & C_{2} 1 & - 2 - 1 & 0 2 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$
$R_{1}, R_{2}, R_{3}$ are rows of $A$ and $C_{1}, C_{2}$ are columns of $B$ .
$∴ A B = {⎡ ⎢ ⎣ \begin{matrix} R_{1} C_{1} & R_{1} C_{2} R_{2} C_{1} & R_{2} C_{2} R_{3} C_{1} & R_{3} C_{2} \end{matrix} ⎤ ⎥ ⎦}_{3 \times 2}$
$= {⎡ ⎢ ⎣ \begin{matrix} 0 \times 1 + 1 \times (- 1) + 2 \times 2 & 0 \times (- 2) + 1 \times 0 + 2 \times (- 1) 1 \times 1 + 2 \times (- 1) + 3 \times 2 & 1 \times (- 2) + 2 \times 0 + 3 \times (- 1) 2 \times 1 + 3 \times (- 1) + 4 \times 2 & 2 \times (- 2) + 3 \times 0 + 4 \times (- 1) \end{matrix} ⎤ ⎥ ⎦}_{3 \times 2}$
$= {⎡ ⎢ ⎣ \begin{matrix} 0 - 1 + 4 & 0 + 0 - 2 1 - 2 + 6 & - 2 + 0 - 3 2 - 3 + 8 & - 4 + 0 - 4 \end{matrix} ⎤ ⎥ ⎦}_{3 \times 2}$
$= {⎡ ⎢ ⎣ \begin{matrix} 3 & - 2 5 & - 5 7 & - 8 \end{matrix} ⎤ ⎥ ⎦}_{3 \times 2}$
Since
the number of columns of $B$ is $2$ and the number of rows of $A$
is $3$ . $B A$ is not defined. $(∵ 2 \neq 3)$

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