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Let $P [\begin{array}{ccc} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array}]$ and I be the identity matrix
of order 3. If $Q = [q_{i i}]$ ] is a matrix such that $P^{50} - Q = I$ then,
$\frac{9_{31} + q_{32}}{q_{21}}$ equal

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a

52

b

103

c

201

d

205

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detailed solution

Correct option is B

We have P=1004101641∴P2=PP=10041016411004101641=10081016+3281P3=P2P=10081016+32811004101641=100121016+32+48121By observing the symmetry, we obtain P50=4×5010 16+32+48+…50 terms 4×501⇒ P50=1002001016×50×5122001 ∴ P50−Q=I⇒ Q=P50−I=00020000204002000⇒ q21=200,q31=20400 and q32=200∴ q31+q32q21=20400+200200=20600200=103

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