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MATRICES — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMATRICES5 Marks
Question
If $\text{A}=\begin{bmatrix}1&2\\-2&1\end{bmatrix},\ \text{B}=\begin{bmatrix}2&3\\3&-4\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&0\\-1&0\end{bmatrix},$ verify $\text{A}(\text{B}+\text{C})=\text{AB}+\text{AC}.$
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Answer

We have, $\text{A}=\begin{bmatrix}1&2\\-2&1\end{bmatrix},\ \text{B}=\begin{bmatrix}2&3\\3&-4\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&0\\-1&0\end{bmatrix}$
$\text{B}+\text{C}=\begin{bmatrix}2&3\\3&-4\end{bmatrix}+\begin{bmatrix}1&0\\-1&0\end{bmatrix}=\begin{bmatrix}3&3\\2&-4\end{bmatrix}$
$\Rightarrow\ \text{A}.(\text{B}+\text{C})=\begin{bmatrix}1&2\\-2&1\end{bmatrix}.\begin{bmatrix}3&3\\2&-4\end{bmatrix}$
$=\begin{bmatrix}3+4&3-8\\-6+2&-6-4\end{bmatrix}$
$=\begin{bmatrix}7&-5\\-4&-10\end{bmatrix}\ ....(\text{i})$
$\text{AB}=\begin{bmatrix}1&2\\-2&1\end{bmatrix}\begin{bmatrix}2&3\\3&-4\end{bmatrix}$
$=\begin{bmatrix}2+6&3-8\\-4+3&-6-4\end{bmatrix}=\begin{bmatrix}8&-5\\-1&-10\end{bmatrix}$
and $\text{AC}=\begin{bmatrix}1&2\\-2&1\end{bmatrix}\begin{bmatrix}1&0\\-1&0\end{bmatrix}$
$=\begin{bmatrix}1-2&0\\-2-1&0\end{bmatrix}=\begin{bmatrix}-1&0\\-3&0\end{bmatrix}$
$\therefore\ \text{AB}+\text{AC}=\begin{bmatrix}8&-5\\-1&-10\end{bmatrix}+\begin{bmatrix}-1&0\\-3&0\end{bmatrix}$
$\Rightarrow\ \text{AB}+\text{AC}=\begin{bmatrix}7&-5\\-4&-10\end{bmatrix}\ ....(\text{ii})$
From Eq. (i) and (ii),
$\text{A}(\text{B}+\text{C})=\text{AB}+\text{AC}.$

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