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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
$8$
$10$
$11$
$12$
Solution
$(\mathrm{I}-\mathrm{A})^{3}=\mathrm{I}^{3}-\mathrm{A}^{3}-3 \mathrm{~A}(\mathrm{I}-\mathrm{A})=\mathrm{I}-\mathrm{A}^{3}$
$\Rightarrow 3 \mathrm{~A}(\mathrm{I}-\mathrm{A})=0 \text { or } \mathrm{A}^{2}=\mathrm{A}$
$\Rightarrow\left[\begin{array}{cc}\mathrm{a}^{2} & \mathrm{ab}+\mathrm{bd} \\ 0 & \mathrm{~d}^{2}\end{array}\right]=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ 0 & \mathrm{~d}\end{array}\right]$
$\Rightarrow \mathrm{a}^{2}=\mathrm{a}, \mathrm{b}(\mathrm{a}+\mathrm{d}-1)=0, \mathrm{~d}^{2}=\mathrm{d}$
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
