Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao a > 0, A₁ = 0, μo = 0, and μ₁ => 0. (a) Show that the infinitesimal generator matrix A = Thus, for n = 1,2,3,..., A" = (-a - b)-1A holds. -a a b satisfies the property A² = (a + b)A. (b) Using the result from part (a) to compute the matrix exponential etA in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim∞ P(t) and use it to find the stationary distribution.

Let A be a 5 x 7 matrix. Which of the following conditions must be true for \(A)? Select all the correct options. rank(A) ≤ 5 ker(A) = {0} A = always have infinitely many solutions. O If rank(A) = 5, then A is invertible.

4. (a) Prove that if A is an n x n invertible matrix that satisfies the equation A2 = A then A is the identity matrix. (b) Write down a 2 x 2 matrix B that is not the identity but satisfies B² = B. %3D

Let P denote a matrix whose columns are eigenvectors K,, K2, ..., K, corresponding to distinct eigenvalues 1,, 1,,..., , of an nxn matrix A. Then it can be shown that A = PDP-1, where D is a diagonal matrix defined by the following. D = (9) …… 入。 If D is defined as in (9), then find e Dt eDt