Please answer in detail. The question is attached below.

Consider the nite difference matrix operator for the 1D model problem d2u(:L')/da:2 : f (x) on domain [0, 1] with boundary conditions 15(0) 2 0 and u(1) = 0, given by 21 A = e an\". This matrix can be considered a discrete version of the continuous operator d2 / (13:2 that acts upon a function u(a:). (a) Show that the n eigenvectors of A are given by the vectors 0\") (19 : 1, . . . ,n) with components as?) = sin(p7rjh), and with eigenvalues 2 AP = W(cos(p7rh) 1). (Hint: verify for a general component (A 531mb, using trigonometric identities.) (b) Verify that the functions u(p)(m) = sin(p7m:) (p = 1, 2, . . .) are eigenfunctions of the continuous differential operator dZ/dzr2 on domain [0, 1] with boundary conditions u(0) : 0 and u(1) = 0. (Here, u(x) is an eigenfunction of 032/011:2 with eigenvalue A if d2u(33)/d:132 = Au(:t), and u(m) satises the boundary conditions.) What are the eigenvalues? Compare the eigenvectors and eigenvalues for the discrete and the continuous operators and comment (we would expect the discrete and continuous eigenvalues and eigenvectors to be similar, if the discrete operator is a useful approximation of the continuous operator ....) Are the discrete and continuous eigenvalues similar for small values of h p