Let P = ⎡⎢⎣1 4 16 0 1 4 0 0 1⎤⎥⎦ and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P50 - Q = I, then q31 + q32q21 equals

P = ⎡⎢⎣1 0 0 4 1 0 16 4 1⎤⎥⎦ ⇒ P2 = ⎡⎢⎣1 0 0 8 1 0 16(1+2) 8 1⎤⎥⎦, P3 = ⎡⎢⎣1 0 0 12 1 0 16(1+2+3) 12 1⎤⎥⎦
P50 = ⎡⎢⎣1 0 0 4×50 1 0 16(1+2+..50) 4×50 1⎤⎥⎦, P50 = ⎡⎢⎣1 0 0 200 1 0 20400 200 1⎤⎥⎦
∵ P50 - Q = I ⇒ ⎡⎢⎣1 0 0 200 1 0 20400 200 1⎤⎥⎦ - ⎡⎢⎣q11 q12 q13 q21 q22 q23 q31 q32 q33⎤⎥⎦ = ⎡⎢⎣1 0 0 0 1 0 0 0 1⎤⎥⎦
⇒ 200 - q21 = 0 ⟹ q21 = 200 ⇒ 20400 - q31 = 0 ⇒ q31 = 20400 and 200 - q32 = 0 ⇒ q32 = 200
∴ q31 + q32q21 = 20400 + 200(200) = 103