Let A = ⎡⎢⎣100110111⎤⎥⎦ and B = A20. Then the sum of the elements of the first column of B is?

Given, A = ⎡⎢⎣100110111⎤⎥⎦. Computing higher powers of A, A2 = A.A = ⎡⎢⎣100110111⎤⎥⎦ × ⎡⎢⎣100110111⎤⎥⎦ = ⎡⎢⎣100210321⎤⎥⎦
A3 = A2.A = ⎡⎢⎣100210321⎤⎥⎦ × ⎡⎢⎣100110111⎤⎥⎦ = ⎡⎢⎣100310631⎤⎥⎦
A4 = A3.A = ⎡⎢⎣100310631⎤⎥⎦ × ⎡⎢⎣100110111⎤⎥⎦ = ⎡⎢⎣1004101041⎤⎥⎦.
On observing the pattern, we come to a conclusion that, Ak = ⎡⎢⎢⎢⎣100k10k(k+1)2k1⎤⎥⎥⎥⎦. Therefore, the sum of the first column of A20 is (1 + 20 + 20 × 21 / 2) = 231. Option C is correct.