1. Consider the following matrix: 2 2 A = 2 1 1 4 (a) Find the...

  

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1. Consider the following matrix: 2 2 A = 2 1 1 4 (a) Find the eigenvalues of the matrix A. Verify that your eigenvalues are correct by using either the trace or the determinant of A. (b) Find the corresponding eigenvectors. (c) Can A be diagonalised? If so, write down the matrix P that will diagonalise A, and write down the corresponding diagonal matrix D. If the matrix cannot be diago- nalised, explain why. (d) Consider the following system of linear first order differential equations: dx = x+2y+2z dt dy = x + 2y - z dt dz = -x+y+4z. dt Write down the general solution to the above system of equations. You may use your answers to the questions above. (4+6+2+2=14 marks) 2. Give an example of a 3 3 matrix A with the following properties: (a) The eigenvalues of A are = 0, A2 = 4, A3 = 4 and (b) The corresponding eigenvectors of A are V1 = (1,1,1), v = (1,0,-1) and V3 = (1, -2, 1). Explain your reasoning. 3. Solve the system x' -3x+6y, y' = -2x+8y, subject to the initial conditions x(0) = 2, y(0) = 1. Show your work. (6 marks) (5 marks) Total: 25 marks 1. Consider the following matrix: 2 2 A = 2 1 1 4 (a) Find the eigenvalues of the matrix A. Verify that your eigenvalues are correct by using either the trace or the determinant of A. (b) Find the corresponding eigenvectors. (c) Can A be diagonalised? If so, write down the matrix P that will diagonalise A, and write down the corresponding diagonal matrix D. If the matrix cannot be diago- nalised, explain why. (d) Consider the following system of linear first order differential equations: dx = x+2y+2z dt dy = x + 2y - z dt dz = -x+y+4z. dt Write down the general solution to the above system of equations. You may use your answers to the questions above. (4+6+2+2=14 marks) 2. Give an example of a 3 3 matrix A with the following properties: (a) The eigenvalues of A are = 0, A2 = 4, A3 = 4 and (b) The corresponding eigenvectors of A are V1 = (1,1,1), v = (1,0,-1) and V3 = (1, -2, 1). Explain your reasoning. 3. Solve the system x' -3x+6y, y' = -2x+8y, subject to the initial conditions x(0) = 2, y(0) = 1. Show your work. (6 marks) (5 marks) Total: 25 marks