Let A = pmatrix 2 & 1 0 & 2 pmatrix and S = _ k=1 ¹⁰ A^k . If the sum of all elements of the matrix S is equal to m 2¹⁰ - p , where m and p are positive integers, then the value of m + p is equal to ______.

Step-by-step solution

We can write A = 2I + N , where I = pmatrix 1 & 0 0 & 1 pmatrix and N = pmatrix 0 & 1 0 & 0 pmatrix . Notice that N^2 = pmatrix 0 & 0 0 & 0 pmatrix . Using the binomial theorem, A^k = (2I + N)^k = 2^k I + k 2^ k-1 N = pmatrix 2^k & k 2^ k-1 0 & 2^k pmatrix . Therefore, S = _ k=1 ¹⁰ A^k = pmatrix _ k=1 ¹⁰ 2^k & _ k=1 ¹⁰ k 2^ k-1 0 & _ k=1 ¹⁰ 2^k pmatrix . Let a = _ k=1 ¹⁰ 2^k = 2^1 + 2^2 + + 2¹⁰ = 2(2¹⁰ - 1) 2 - 1 = 2¹¹ - 2 . Let b = _ k=1 ¹⁰ k 2^ k-1 = 1 2^0 + 2 2^1 + 3 2^2 + + 10 2^9 . Multiply by 2 : 2b = 1 2^1 +