Question
समुच्चय $\left\{ A =\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right): a , b , d \in\{-1,0,1\}\right.$ तथा $\left.( I - A )^{3}= I - A ^{3}\right\}$ I, $2 \times 2$ का तत्समक आव्यूह है में अवयवों की संख्या है
$\Rightarrow 3 \mathrm{~A}(\mathrm{I}-\mathrm{A})=0 \text { or } \mathrm{A}^{2}=\mathrm{A}$
$\Rightarrow\left[\begin{array}{cc}\mathrm{a}^{2} & \mathrm{ab}+\mathrm{bd} \\ 0 & \mathrm{~d}^{2}\end{array}\right]=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ 0 & \mathrm{~d}\end{array}\right]$
$\Rightarrow \mathrm{a}^{2}=\mathrm{a}, \mathrm{b}(\mathrm{a}+\mathrm{d}-1)=0, \mathrm{~d}^{2}=\mathrm{d}$
If $b \neq 0, a+d=1 \Rightarrow 4$ ways
If $\mathrm{b}=0, \mathrm{a}=0,1\;and\; \mathrm{~d}=0,1 \Rightarrow 4$ ways
$\Rightarrow$ Total $8$ matrices
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$(A)$ $y\left(\frac{\pi}{4}\right)=\frac{\pi^2}{8 \sqrt{2}}$
$(B)$ $y^{\prime}\left(\frac{\pi}{4}\right)=\frac{\pi^2}{18}$
$(C)$ $y\left(\frac{\pi}{3}\right)=\frac{\pi^2}{9}$
$(D)$ $y ^{\prime}\left(\frac{\pi}{3}\right)=\frac{4 \pi}{3}+\frac{2 \pi^2}{3 \sqrt{3}}$
$(A)$ $\vec{b}=(\vec{b} \cdot \vec{z})(\vec{z}-\vec{x})$
$(B)$ $\vec{a}=(\vec{a} \cdot \vec{y})(\vec{y}-\vec{z})$
$(C)$ $\vec{a} \cdot \vec{b}=-(\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z})$
$(D)$ $\vec{a}=(\vec{a} \cdot \vec{y})(\vec{z}-\vec{y})$