MATH 2050 Linear Algebra

(Section 4)
2009 Winter

Assignment 3   -   Solutions

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

    Matrices A and B have the same dimensions, 3×2.
    The expression is therefore defined.
    A + 2B = ...

    A + 2B = [ 1 12 ; 0 7 ; 1 6 ]

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

    Matrix C has dimensions 2×3 and matrix D has dimensions 3×2
    implies Matrices CT and D have the same dimensions.
    The expression is therefore defined.
    2C' - 3D = ...

    (2C - 3D')' = [ 2 3 ; 10 -5 ; 15 -10 ]

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

    Matrix C has dimensions 2×3 but matrix D has dimensions 3×2     implies

      C + D   is undefined.  

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

    Matrix D has dimensions 3×2 (so that its transpose DT has dimensions 2×3) and matrix E has dimensions 2×2
    implies DE is defined (as a 3×2 matrix), EDT is defined (as a 2×3 matrix), but ED is not defined at all.

    DE = [ 1 2 ; 3 8 ; 4 11 ]

    DE = [ 1 2 ; 3 8 ; 4 11 ]

      ED   is undefined.  

    ED' = [ -1 -3 -4 ; 2 4 5 ]

    ED' = [ -1 -3 -4 ; 2 4 5 ]

  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 ]

    MU = [ a (a+b) ; c (c+d) ]
    UM = [ (a+c) (b+d) ; c d ]
    MU = UM
    leads to c = 0 and d = a
    Therefore the most general matrix M that commutes with U is

    M = [ a b ; 0 a ]

    (where a and b are free parameters).

  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.

    R(theta) R(phi) = ...
    R(theta) R(phi) = ...
    R(theta) R(phi) = R(theta + phi)
    But the addition of scalars is commutative
    R(theta) R(phi) = R(phi) R(theta)
    Therefore the two rotation matrices commute.   Reversing the order in which the rotations are carried out has no effect on the end result: a net rotation through an angle of (q + f).

  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)

    Rz(theta) Rx(phi) = ...
    Rz(theta) Rx(phi) = ...
    Rx(phi) Rz(theta) = ...
    Rx(phi) Rz(theta) not= Rz(theta) Rx(phi)
    (unless q = 0 and/or f = 0 , which correspond to the identity operations of no rotation at all about the appropriate axis).   Therefore reversing the order in which non-trivial rotations about each of the x and z axes are taken does result in a different transformation.

  1. Prove that the square K 2 of any skew-symmetric matrix K is symmetric.

    Matrix K is skew-symmetric     (transpose of K) = (negative of K)
    leads to (K^2)T = +K^2
    Therefore the square of any skew-symmetric matrix is a symmetric matrix.

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      Created 2009 01 06 and most recently modified 2009 01 07 by Dr. G.H. George