Q31 / 90 · KVPY 2011 (SB/SX) · Mathematics

Let $P$ be an $m \times m$ matrix such that $P^{2}=P$. Then $(1+P)^{n}$ equals -

  • A
    $I+P$
  • B
    $I+n P$
  • C
    $I+2^{n} P$
  • Correct answer
    D
    $I+\left(2^{n}-1\right) P$

Solution

$\begin{array}{l}\mathrm{P}^{2}=\mathrm{P} \\ \mathrm{P}^{-1} \mathrm{P}^{2}=\mathrm{P}^{-1} \mathrm{P} \\ \mathrm{P}=\mathrm{I} \\ (\mathrm{I}+\mathrm{P})^{\mathrm{n}}=(2 \mathrm{P})^{\mathrm{n}}=2^{\mathrm{n}} \mathrm{P}^{\mathrm{n}}\end{array}$ $=2^{n} \mathrm{P}$ $=\mathrm{P}+\left(2^{\mathrm{n}}-1\right) \mathrm{P}$ $=\mathrm{I}+\left(2^{\mathrm{n}}-1\right) \mathrm{P}$

Hence, the correct answer is option: 'D'.

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