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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES3 Marks
Question
Let $\text{A}=\begin{bmatrix}1&2\\-1&3\end{bmatrix},\ \text{B}=\begin{bmatrix}4&0\\1&5\end{bmatrix},$ $\text{C}=\begin{bmatrix}2&0\\1&-2\end{bmatrix},$ a = 4, b = -2, then show that $\text{A}(\text{BC})=(\text{AB})\text{C}.$
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Answer

We have, $\text{A}=\begin{bmatrix}1&2\\-1&3\end{bmatrix},\ \text{B}=\begin{bmatrix}4&0\\1&5\end{bmatrix},$ $\text{C}=\begin{bmatrix}2&0\\1&-2\end{bmatrix},$ and a = 4, b = -2$(\text{BC})=\begin{bmatrix}4&0\\1&5\end{bmatrix}\begin{bmatrix}2&0\\1&-2\end{bmatrix}$
$=\begin{bmatrix}8&0\\7&-10\end{bmatrix}$
and $\text{A}(\text{BC})=\begin{bmatrix}1&2\\-1&3\end{bmatrix}\begin{bmatrix}8&0\\7&-10\end{bmatrix}$ $=\begin{bmatrix}8+14&0-20\\-8+21&0-30\end{bmatrix}=\begin{bmatrix}22&-20\\13&-30\end{bmatrix}$Also, $(\text{AB})=\begin{bmatrix}1&2\\-1&3\end{bmatrix}\begin{bmatrix}4&0\\1&5\end{bmatrix}$
$=\begin{bmatrix}6&10\\-1&15\end{bmatrix}$
$(\text{AB})\text{C}=\begin{bmatrix}6&10\\-1&15\end{bmatrix}\begin{bmatrix}2&0\\1&-2\end{bmatrix}$
$=\begin{bmatrix}22&-20\\13&-30\end{bmatrix}=\text{A}(\text{BC})$
Hence proved.

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