MATH 2050 Linear Algebra

(Section 4)
2009 Winter

Assignment 3   -   Questions

[Sections 2.1 - 2.2   Matrix Algebra]
Due in class on 2009 February 02 (Monday)

  1. A + 2B , where   A = [ 1 0 ; 2 1 ; 5 -2 ] B = [ 0 6 ; -1 3 ; -2 4 ]         [10 marks]

  1. (2C – 3DT)T , where
    C = [ 1 2 3 ; 3 2 1 ] D = [ 0 1 ; -2 3 ; -3 4 ]         [10 marks]

  1. C + D , where   C = [ 1 2 3 ; 3 2 1 ] D = [ 0 1 ; -2 3 ; -3 4 ]         [10 marks]

  1.   DE, ED and EDT, where
    D = [ 0 1 ; -2 3 ; -3 4 ] E = [ -1 0 ; 1 2 ]         [15 marks]

  1. Find the conditions on the entries a, b, c, d of the matrix M such that it commutes with the upper triangular matrix U, where
    M = [ a b ; c d ] U = [ 1 1 ; 0 1 ]         [15 marks]

  1. The matrix that represents a rotation of the x-y plane by an angle q about the origin is   R(t) = [ cos t -sin t ; sin t cos t ]
    Show that any two such matrices commute, that is
    R(theta) R(phi) = R(phi) R(theta) for all theta and phi
    Therefore the order of rotations in two dimensions doesn’t matter.         [10 marks]

  1. In three dimensions, the matrix that represents a rotation by an angle q about the z axis is
    Rz(t) = [ cos t -sin t 0 ; sin t cos t 0 ; 0 0 1 ]
    and the matrix that represents a rotation by an angle f about the x axis is
    Rx(f) = [ 1 0 0 ; 0 cos f -sin f ; 0 sin f cos f ]
    Show that the order of these two rotations does matter in general, that is
    Rz(theta) Rx(phi) not= Rx(phi) Rz(theta)         [20 marks]

  1. Prove that the square K 2 of any skew-symmetric matrix K is symmetric.         [10 marks]

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      Created 2009 01 05 and most recently modified 2009 01 06 by Dr. G.H. George