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Provide solution for rd sharma maths class 12 chapter adjoint and inverse of a matrix exercise multiple choice question question 11

Answers (1)

option (d) none of these

Given: A^{5}=0 \text { such that } A^{n} \neq \mathrm{I} \text { for } 1 \leq n \leq 4

Hint: zero matrix is that whose every element is zero

Solution:

\begin{aligned} &\mathrm{I}-\mathrm{A}^{5}=(\mathrm{I}-\mathrm{A})\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\mathrm{A}^{4}\right) \\ &\mathrm{I}=(\mathrm{I}-\mathrm{A})\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\mathrm{A}^{4}\right) \quad\left(\therefore \mathrm{A}^{5}=0\right) \\ &\frac{I}{(I-A)}=\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\mathrm{A}^{4}\right) \\ &(\mathrm{I}-\mathrm{A})^{-1}=\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\mathrm{A}^{4}\right) \end{aligned}

Hence, option (d) is correct

 

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