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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 103P=⎡⎢⎣1004101641⎤⎥⎦ ⇒P2=⎡⎢⎣1004×21016(1+2)4×21⎤⎥⎦ ⇒P3=⎡⎢⎣1004×31016(1+2+3)4×31⎤⎥⎦ ⋮ ⋮ Similarly P50=⎡⎢⎣1004×501016(1+2+3+...+50)4×501⎤⎥⎦ ⇒P50=⎡⎢⎣10020010204002001⎤⎥⎦ As P50−Q=I, ∴⎡⎢⎣10020010204002001⎤⎥⎦−⎡⎢⎣q11q12q13q21q22q23q31q32q33⎤⎥⎦=⎡⎢⎣100010001⎤⎥⎦ 20400−q31=0⇒q31=20400 200−q32=0⇒q32=200 200−q21=0⇒q21=200 ∴q31+q32q21=20400+200200=103

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