Let A = bmatrix 1 & 0 & 0 2 & 1 & 0 3 & 2 & 1 bmatrix and let S = _ r=1 ²⁰ A^r . The sum of the elements in the first column of the matrix S is :

Step-by-step solution

Let A = I + N , where I is the identity matrix and N = bmatrix 0 & 0 & 0 2 & 0 & 0 3 & 2 & 0 bmatrix . Calculating the powers of N : N^2 = bmatrix 0 & 0 & 0 2 & 0 & 0 3 & 2 & 0 bmatrix bmatrix 0 & 0 & 0 2 & 0 & 0 3 & 2 & 0 bmatrix = bmatrix 0 & 0 & 0 0 & 0 & 0 4 & 0 & 0 bmatrix N^3 = bmatrix 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 bmatrix Using the binomial expansion for A^r : A^r = (I + N)^r = I + rN + r(r-1) 2 N^2 Summing this over r from 1 to 20 : S = _ r=1 ²⁰ A^r = _ r=1 ²⁰ ( I + rN + r(r-1) 2 N^2 ) S = 20I + ( _ r=1 ²⁰