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Question

# Let P=⎡⎢⎣1004101641⎤⎥⎦ and I be the identity matrix of order 3. If Q=[qij] is a matrix, such that P50−Q=I, then q31+q32q21 equals

A
52
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B
103
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C
201
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D
205
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Solution

## The correct option is B 103Here, P=⎡⎢⎣1004101641⎤⎥⎦ ∴ P2=⎡⎢⎣1004101641⎤⎥⎦⎡⎢⎣1004101641⎤⎥⎦=⎡⎢⎣1004+41016+324+41⎤⎥⎦=⎡⎢⎣1004×21016(1+2)4×21⎤⎥⎦ . . . .(i) and P3=⎡⎢⎣1004×21016(1+2)4×21⎤⎥⎦⎡⎢⎣1004101641⎤⎥⎦=⎡⎢⎣1004×31016(1+2+3)4×31⎤⎥⎦ . . . (ii) From symmetry, P50=⎡⎢⎣1004×501016(1+2+3+⋯+50)4×501⎤⎥⎦ ∵ P50 - Q = I [given] ∴ ⎡⎢ ⎢⎣1−q11−q12−q13200−q211−q22−q2316×502(51)−q31200−q321−q33⎤⎥ ⎥⎦=⎡⎢⎣100010001⎤⎥⎦ ⇒ 200−q21=0, 16×50×512−q31=0,200−q32=0∴ q21=200, q32=200, q31=20400Thus, q31+q32q21=20400+200200=20600200=103

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