MA-305 Homework 4

Due at 4:05pm, Tuesday, October 7, 1997


You may do the calculations necessary for these problems either by hand or with Maple. Please submit a handwritten solution, or email an ASCII/Postscript/html document to the TA. (You may also submit through email which attached your Maple worksheet.)

  1. *Show that det(A) = (b-a)(c-a)(b-c) if A is the following.

    [ a2 a 1 ]
    A =[ b2 b 1 ]
    [ c2 c 1 ]

    This determinant is call a Vandermonde determinant.

    Solution:

    det(A) = a2b - a2c - b2a + b2c + c2a - c2b = (b-a)(c-a)(b-c)

  2. [ 1 -1 2 ]
    *If A =[ 3 4 1 ] , verify that det(A) = det(AT).
    [ 2 5 1 ]

    Solution:

    [ 1 3 2 ]
    AT = [ -1 4 5 ] , det(A) = det(AT) = 14.
    [ 2 1 1 ]

  3. *Verify that det(AB) = det(A)det(B) for the following:

    [ 1 -2 3 ] [ 1 0 2 ]
    A =[ -2 3 1 ] , B = [ 3 -2 5 ]
    [ 0 1 0 ] [ 2 1 3 ]

    Solution:

    det(AB) = -21
    det(A)det(B) = (-7)(3)= -21
    det(AB) = det(A)det(B).

  4. *Is det(AB) = det(BA)? Justify your answer.

    Solution:

    Yes, det(AB) = det(A)det(B) = det(B)det(A) = det(BA).

*: problem from ``Introductory Linear Algebra with Applications'' by Kolman and Hill