Maths with David

    9231. FP2. Further Matrices. Past Exam Questions

    9231/21/M/J/25q8

    (a) It is given that \lambda is an eigenvalue of the non-singular square matrix A, with corresponding eigenvector e. Show that e is an eigenvector of A3, with corresponding eigenvalue \lambda ^3 [2 marks]

    The matrix A is given by: \begin{pmatrix} -1 & 3 & 4 \\ 0 & 1 & 0 \\ 0 & -2 & 5 \end{pmatrix} [2 marks]

    (b) Show that the eigenvalues of A are -1, 1 and 5 [2 marks]

    (c) Find a matrix P and a diagonal matrix D such that A – 2I = PDP-1. [6 marks]

    (d) Use the characteristic equation of A to show that (A – 2I)3 = aA2 + bA + cI, where a, b and c are constants to be determined. [3 marks]