MCQ
The number of elements in the set $\left\{A=\left(\begin{array}{ll}a & b \\ 0 & d\end{array}\right): a, b, d \in\{-1,0,1\}\right.$ and $\left.(I-A)^{3}=I-A^{3}\right\}$ where $I$ is $2 \times 2$ identity matrix, is :
- ✓$8$
- B$10$
- C$11$
- D$12$
$\Rightarrow 3 \mathrm{~A}(\mathrm{I}-\mathrm{A})=0 \text { or } \mathrm{A}^{2}=\mathrm{A}$
$\Rightarrow\left[\begin{array}{cc}\mathrm{a}^{2} & \mathrm{ab}+\mathrm{bd} \\ 0 & \mathrm{~d}^{2}\end{array}\right]=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ 0 & \mathrm{~d}\end{array}\right]$
$\Rightarrow \mathrm{a}^{2}=\mathrm{a}, \mathrm{b}(\mathrm{a}+\mathrm{d}-1)=0, \mathrm{~d}^{2}=\mathrm{d}$
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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| $X$ | 2 | 3 | 4 | 5 |
| $P(X)$ | $\frac{5}{k}$ | $\frac{7}{k}$ | $\frac{9}{k}$ | $\frac{11}{k}$ |