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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{P}(\text{x})=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix},$ then show that P(x).P(y) = P(x + y) = P(y)P(x).
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Answer

We have, $\text{P}(\text{x})=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix}$ $\Rightarrow\ \text{P}(\text{y})=\begin{bmatrix}\cos\text{y}&\sin\text{y}\\-\sin\text{y}&\cos\text{y}\end{bmatrix}$ Consider, P(x).P(y) = P(x + y)LHS = P(x).P(y)
$=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix}\begin{bmatrix}\cos\text{y}&\sin\text{y}\\-\sin\text{y}&\cos\text{y}\end{bmatrix}$
$=\begin{bmatrix}\cos\text{x}.\cos\text{y}-\sin\text{x}.\sin\text{y}&\cos\text{x}.\sin\text{y}+\sin\text{x}.\cos\text{y}\\-\sin\text{x}.\cos\text{y}-\cos\text{x}.\sin\text{y}&-\sin\text{x}.\sin\text{y}+\cos\text{x}.\sin\text{y}\end{bmatrix}$
$=\begin{bmatrix}\cos(\text{x}+\text{y})&\sin(\text{x}+\text{y})\\-\sin(\text{x}+\text{y})&\cos(\text{x}+\text{y})\end{bmatrix}$
$\begin{bmatrix}\because\ \cos(\text{x}+\text{y})=\cos\text{x}.\cos\text{y}-\sin\text{x}.\sin\text{y}\\\text{and }\sin(\text{x}+\text{y})=\sin\text{x}.\cos\text{y}+\cos\text{x}.\sin\text{y}\end{bmatrix}$
$=\text{P}(\text{x}+\text{y})$
= RHS Hence proved.

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