MXB106 Take Home Assignment 2


MXB106 Take Home Assignment 2

Question 1 Let A be a n × n matrix. Prove that the coefficient of λ n in the characteristic polynomial is 1. In other words, show that the characteristic polynomial is monic. Note that you are not required to find the eigenvalues of A, just a general expression for the characteristic polynomial.

Question 2 Show that the matrix          0 0 0 · · · 0 −c0 1 0 0 · · · 0 −c1 0 1 0 · · · 0 −c2 0 0 1 · · · 0 −c3 . . . . . . . . . . . . . . . 0 0 0 · · · 1 −cn−1          has the characteristic polynomial p(λ) = c0 + c1λ + . . . + cn−1λ n−1 + λ n Hint: if you are stuck, try solving the problem with a 3 × 3 matrix and working your way up to a general n × n.

Question 3 If A is an n × n matrix then prove that det(adj(A)) = det(A) n−1

Question 4 For which real values of α do the following vectors ~v1 = (α, −1/2, −1/2) ~v2 = (−1/2, α, −1/2) ~v3 = (−1/2, −1/2, α) form a linearly dependent set in R3 1

Question 5 Show that if 0 < θ < π then A = cos(θ) − sin(θ) sin(θ) cos(θ) has no real eigenvalues and consequently no real eigenvectors.

Question 6 Find the eigenvalues of A =        c1 c1 c1 · · · c1 c2 c2 c2 · · · c2 c3 c3 c3 · · · c3 . . . . . . . . . . . . cn cn cn · · · cn        Hint: You may find that the identity A 2 = (c1 + c2 + . . . + cn)A is useful. Question 7 Use determinants to show that the equation of a straight line passing through the distinct point (a1, b1) and (a2, b2) can be written as x y 1 a1 b1 1 a2 b2 1 = 0 Question 8 Let A ∈ R2×2 and D ∈ R2×2 . That is A = a1,1 a1,2 a2,1 a2,2 , D = d1,1 d1,2 d2,1 d2,2 Show that det A 0 0 D = det(A) det(D) 2 Question 9 Let A =   3 −1 −2 2 0 −2 2 −1 −1   Find the eigenvalues of A and the corresponding eigenvectors. State the multiplicity of each eigenvalue. Question 10 Following question 9, find a diagonmal matrix D, and a matrix P where P ∈ R3×3 which satisfy A = PDP−1 3

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