Let A be an n × n matrix of the following form. What is the value of the…

Let D_n denote the determinant of the n x n tridiagonal matrix with 3 on the main diagonal and 1 on the two adjacent diagonals. Expanding along the first row gives the recurrence D_n = 3D_{n-1} - D_{n-2}, with D_1 = 3 and D_2 = 8. The characteristic equation is r^2 - 3r + 1 = 0, so r = (3 + sqrt(5))/2 or (3 - sqrt(5))/2. Therefore D_n = c1((3 + sqrt(5))/2)^(n-1) + c2((3 - sqrt(5))/2)^(n-1). Using D_1 = 3 and D_2 = 8 gives c1 = (3sqrt(5) + 7)/(2sqrt(5)) and c2 = (3sqrt(5) - 7)/(2sqrt(5)). This is exactly the expression in option D.