Let P=100410841 and I be the identity matrix of order 3. If Q=[qij] is a matrix such that  p500−Q=I, thenq31+q32q21 equals

# Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 8& 4& 1\end{array}\right]$ and I be the identity matrix of order 3. If $\mathrm{Q}=\left[{\mathrm{q}}_{\mathrm{ij}}\right]$ is a matrix such that

1. A

995

2. B

1001

3. C

1000

4. D

1005

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+91

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### Solution:

$\mathrm{P}=\mathrm{I}+\mathrm{B},\mathrm{B}=\left[\begin{array}{lll}0& 0& 0\\ 4& 0& 0\\ 8& 4& 0\end{array}\right]$

${\mathrm{B}}^{2}=\left[\begin{array}{lll}0& 0& 0\\ 4& 0& 0\\ 8& 4& 0\end{array}\right]\left[\begin{array}{lll}0& 0& 0\\ 4& 0& 0\\ 8& 4& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 16& 0& 0\end{array}\right]$

${\mathrm{B}}^{3}=0$

${\mathrm{P}}^{500}=\left(\mathrm{I}+\mathrm{B}{\right)}^{500}=\mathrm{I}+500\mathrm{B}+250×499{\mathrm{B}}^{2}$

$\frac{500×8+250×499×16+500×4}{500×4+0}=1001$  +91

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