For x > 0 , let A = x + 1 x 0 0 0 1 x 0 0 0 12 , B = x 6 x 2 + 1 0 0 0 x 4 0 0 0 1 36 be two matrices and C = A B + A B 2 + … + A B n . Then, T r l i m n → ∞ C is equal to (Where T r A is the trace of the matrix A i.e. the sum of the principal diagonal elements of A )

Step-by-step solution

A B = 1 6 0 0 0 1 4 0 0 0 1 3 A B 2 = 1 6 2 0 0 0 1 4 2 0 0 0 1 3 2 C = 1 6 + 1 6 2 + … + 1 6 n 0 0 0 1 4 + 1 4 2 + … + 1 4 n 0 0 0 1 3 + 1 3 2 + … + 1 3 n l i m n → ∞ C = 1 5 0 0 0 1 3 0 0 0 1 2 T r l i m n → ∞ C = 1 5 + 1 3 + 1 2 = 6 + 10 + 15 30