Let A = pmatrix 1 & a 0 & 2 pmatrix , where a 1, 2, 3, , 100 . Let B = _ k=1 ¹⁰ A^k . The number of values of a such that the sum of all elements of the matrix B is divisible by 7 is _____.

Step-by-step solution

Given A = pmatrix 1 & a 0 & 2 pmatrix . Let us find the general form of A^k . A^2 = pmatrix 1 & a 0 & 2 pmatrix pmatrix 1 & a 0 & 2 pmatrix = pmatrix 1 & a + 2a 0 & 4 pmatrix = pmatrix 1 & 3a 0 & 2^2 pmatrix A^3 = pmatrix 1 & 3a 0 & 4 pmatrix pmatrix 1 & a 0 & 2 pmatrix = pmatrix 1 & a + 6a 0 & 8 pmatrix = pmatrix 1 & 7a 0 & 2^3 pmatrix Observing the pattern, A^k = pmatrix 1 & a(2^k - 1) 0 & 2^k pmatrix . Now, B = _ k=1 ¹⁰ A^k = pmatrix _ k=1 ¹⁰ 1 & _ k=1 ¹⁰ a(2^k - 1) 0 & _ k=1 ¹⁰ 2^k pmatrix . Let us evaluate the