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1MA01: Mathematical Methods Tutorial Sheet 1 Remember, to find the eigenvalues (λ) you use the characteristic equation of a matrix A: det(A − λI) = 0 and then for each λ solve Av = λv for the eigenvector v. 1. Given the matrix A= 3 6 1 4 ! . (1) Find its eigenvalues and corresponding eigenvectors. 2. Find the eigenvalues and eigenvectors of the matrix A= 0 1 −2 −3 ! . (2) 3. A square triangular matrix has all zeros either above or below the main diagonal. The matrices below are all examples of this. Compute their determinants in the usual way and notice that for these special matrices det(A) = a11 a22 . . . ann ie the determinant is just the product of the entries on the diagonal! 5 0 0 A= 0 −3 0 ; B = 0 0 4 6 0 2 −1 ! 4. Consider the matrix A= 6 0 9 −4 10 5 1 ; C= 0 0 −4 (3) 0 0 6 ! (4) Determine its eigenvalues. What do you notice about the eigenvalues when compared with the matrix entries? 1 Sinéad Ryan, ryan@maths.tcd.ie, see also http://www.maths.tcd.ie/˜ryan/123.html 5. Use the Cayley Hamilton theorem to determine A5 given, A= 3 1 2 4 ! (5) 6. A Leslie matrix is given by L= 0.728 1.302 0.52 0.62 ! (6) Find the long-term (percentage) growth rate and the stable age population.