1. (a) i. Calculate the determinant of the following matrices 3 NO 0 A = D = 0 10 14 ii. For the four matrices above compute their kernel; write a basis for the kernel and state its dimension. iii. Notice that the determinant of a matrix is zero whenever the kernel is non-trivial. Can you explain why these two properties are equivalent? (b) A non-zero vector v is an eigenvector for a square matrix A if Av = Av for some A E R. Why is A being an eigenvector of A equivalent to det(A - XI) = 0? (c) Compute the eigenvalues and corresponding eigenvectors of 9 -7 3 M = 3 (1) 16 -16 Show that M can be diagonalised; that is, show that M = CDC-1 where D is diagonal and C some matrix you must specify.2. The matrix exponential is dened by X " EXPO!) = E F 1:20 for X a square matrix and where X to the power zero is equal to the identity matrix. (3) Compute X" for when X is equal to each of the following matrices ar k = 1, 2, 3, . . . 3(1):} (11) (11) (01) 0 U 0 0 l 1 1 1 0 (b) Hence, nd the exponential of each of the four matrices above. (c) Find exp(M) for M given above in (1). 3. Let A be a real, square matrix. Suppose it admits a complex eigenvalue A = a. + ib, where 6,!) E R. (a) By making reference to the characteristic polynomial of A, explain why X = a ib must also be an eigenvalue. Furthermore, explain why the corresponding eigenvector {c of A is necessarily complex. (b) Decompose the eigenvector a: into real and imaginary parts as a: = u +iv. Show that y = u v is an eigenvector for A. (c) Suppose from now on that A is a 2 x 2-matrix and that u and 'U are independent. \"Fith respect to the basis {11,11} Show that A is a. b b a ' (d) The differential equation 2 = Az admits two independent complex solutions 2103) = eA'z, and a0) = ex'y. By splitting these complex solutions into their real and imaginary parts obtain two independent real solutions to the differential equation. Your solutions should be written in terms of u, t! and tab