Let A = bmatrix 2 & 1 0 & 2 bmatrix and B = _ k=1 ¹⁰ A^k . The value of the element in the first row and second column of the matrix B is equal to

Step-by-step solution

We can write the matrix A as A = 2I + N , where I = bmatrix 1 & 0 0 & 1 bmatrix and N = bmatrix 0 & 1 0 & 0 bmatrix . Note that N^2 = bmatrix 0 & 0 0 & 0 bmatrix . Since 2I and N commute, we can use the binomial expansion to find A^k : A^k = (2I + N)^k = (2I)^k + k(2I)^ k-1 N + Because N^2 = 0 , all higher terms vanish. A^k = 2^k I + k 2^ k-1 N Substituting the matrices: A^k = bmatrix 2^k & 0 0 & 2^k bmatrix + bmatrix 0 & k 2^ k-1 0 & 0 bmatrix = bmatrix 2^k & k 2^ k-1 0 & 2^k bmatrix The element in the first row a