Show that AB in the following cases: A= 5&-1 6&7 and B= 2&1 3&4 - Vidyadip

Question

Show that $\text{AB}\neq\text{BA}$ in the following cases:
$\text{A}=\begin{bmatrix}5&-1\\6&7\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2&1\\3&4\end{bmatrix}$

✓

Answer

$\text{AB}=\begin{bmatrix}5&-1\\6&7\end{bmatrix}\begin{bmatrix}2&1\\3&4\end{bmatrix}$
$\Rightarrow\text{AB}=\begin{bmatrix}10-3&5-4\\12+21&6+28\end{bmatrix}$
$\Rightarrow\text{AB}=\begin{bmatrix}7&1\\33&34\end{bmatrix}\ \dots(1)$
Also,
$\text{BA}=\begin{bmatrix}2&1\\3&4\end{bmatrix}\begin{bmatrix}5&-1\\6&7\end{bmatrix}$
$\Rightarrow\text{BA}=\begin{bmatrix}10+6&-2+7\\15+24&-3+28\end{bmatrix}$
$\Rightarrow\text{BA}=\begin{bmatrix}16&5\\39&25\end{bmatrix}\ \dots(2)$
$\therefore\ \text{AB}\neq\text{BA}$ From eqs. (1) and (2)

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