Let A = bmatrix 1 & 1 & 1 0 & 1 & 1 0 & 0 & 1 bmatrix and B = _ k=1 ²⁰ A^k . The sum of all the elements in the first row of the matrix B is equal to _______.

Step-by-step solution

Let A = I + N , where I is the 3 3 identity matrix and N = bmatrix 0 & 1 & 1 0 & 0 & 1 0 & 0 & 0 bmatrix . Calculating the powers of N , we get: N^2 = bmatrix 0 & 0 & 1 0 & 0 & 0 0 & 0 & 0 bmatrix and N^3 = O . Using the binomial expansion, the k^ th power of A is: A^k = (I + N)^k = I + kN + k(k-1) 2 N^2 A^k = bmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 bmatrix + bmatrix 0 & k & k 0 & 0 & k 0 & 0 & 0 bmatrix + bmatrix 0 & 0 & k(k-1) 2 0 & 0 & 0 0 & 0 & 0 bmatrix A^k = bmatrix 1 & k & k + k(k-1) 2 0 & 1 & k 0 & 0 & 1 bmat