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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
For the matrices A and B, verify that (AB)' = B'A' where
  1. $\text{A}=\begin{bmatrix}1\\-4\\3\end{bmatrix},\text{B}=\begin{bmatrix}-1&2&1\end{bmatrix}$
  2. $\text{A}=\begin{bmatrix}0\\1\\2\end{bmatrix},\text{B}=\begin{bmatrix}1&5&7\end{bmatrix}$
✓

Answer

  1. $\text{AB}= \begin{bmatrix}1\\-4\\3\end{bmatrix}\begin{bmatrix}-1&2&1\end{bmatrix}=\begin{bmatrix}-1&2&1\\4&-8&-4\\-3&6&3\end{bmatrix}$
$\therefore\ \text{(AB)}'=\begin{bmatrix}-1&4&-3\\2&-8&6\\1&-4&3\end{bmatrix}$

Now, $\text{A}'=\begin{bmatrix}1&-4&3\end{bmatrix},\text{B}'=\begin{bmatrix}-1\\2\\1\end{bmatrix}$

$\therefore\ \text{B}'\text{A}'=\begin{bmatrix}-1&\\2\\1\end{bmatrix}\begin{bmatrix}1&-4&3\end{bmatrix}=\begin{bmatrix}-1&4&-3\\2&-8&6\\1&-4&3\end{bmatrix} $

Hence, we have verified that (AB)' = B'A'.
  1. $\text{AB}=\begin{bmatrix}0\\1\\2\end{bmatrix}\begin{bmatrix}1&5&7\end{bmatrix}=\begin{bmatrix}0&0&0\\1&5&7\\2&10&14\end{bmatrix}$
$\therefore\ \text{(AB)}'=\begin{bmatrix}0&1&2\\0&5&10\\0&7&14\end{bmatrix}$

Now, $\text{A}'=\begin{bmatrix}0&1&2\end{bmatrix},\text{B}'=\begin{bmatrix}1\\5\\7\end{bmatrix}$

$\therefore\ \text{B}'\text{A}'=\begin{bmatrix}1\\5\\7\end{bmatrix}\begin{bmatrix}0&1&2\end{bmatrix}=\begin{bmatrix}0&1&2\\0&5&10\\0&7&14\end{bmatrix}$

Hence, we have verified that (AB)' = B'A'.

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