If A = 1 1 1 0 1 1 0 0 1 and M = A + A 2 + A 3 + … + A 20 , then the sum of all the elements of the matrix M is equal to _______.

Step-by-step solution

We have, A = 1 1 1 0 1 1 0 0 1 and M = A + A 2 + A 3 + … + A 20 Now, A 2 = A . A = 1 1 1 0 1 1 0 0 1 1 1 1 0 1 1 0 0 1 = 1 2 3 0 1 2 0 0 1 ⇒ A 3 = A 2 . A = 1 1 1 0 1 1 0 0 1 1 2 3 0 1 2 0 0 1 = 1 3 6 0 1 3 0 0 1 . . . . A n = 1 n n n + 1 2 0 1 n 0 0 1 So, required sum = 20 × 3 + 2 ∑ r r = 1 20 + ∑ r = 1 20 r r + 1 2 = 20 × 3 + 2 ∑ r r = 1 20 + 1 2 ∑ r = 1 20 r 2 + 1 2 ∑ r = 1 20 r = 20 × 3 + 5 2 ∑ r = 1 20 r + 1 2 ∑ r = 1 20 r 2 = 20 × 3 + 5