MA-305 Homework 3

Due at 2:35 pm, Thursday, February 26, 1998


Calculations necessary for these problems may be done either by hand or with Maple. Solutions may be submitted in person in class, or you may email an ASCII text or Maple Worksheet (.mws) to the TA, John Haws (jchaws@eos.ncsu.edu).

  1. Find the inverse of the following matrix:

       [ 1 1 0 3 ]
   [ 0 2 0 2 ]
   [ 0 0 2 5 ]
   [ 0 0 0 1 ]

  2. Find the inverse of the following matrix, where a and b are non-zero.

       [ 1 a 0 0 ]
   [ a 1 b 0 ]
   [ 0 b 1 a ]
   [ 0 0 a 1 ]

  3. Find matrices T and U (U row-echelon form) such that TA = U, for

    A = 
       [   2  -2    2 ]
   [ -12   6   -1 ]
   [  -6  -2  -10 ]

  4. Find T-1 for the matrix T in Problem 3 above.

  5. Suppose A is a 3x3 matrix with LU factors

    [ 1
[ l2,1
[ l3,1
     0
 1
 l3,2
     0 ]
 0 ]
 1 ]
    [ u1,1
[ 0
[ 0
     u1,2
 u2,2
 0
     u1,3 ]
 u2,3 ]
 u3,3 ]
    Find det(A).

  6. For
    A = 
       [ a1,1 a1,2 a1,3 ]
   [ a2,1 a2,2 a2,3 ]
   [ a3,1 a3,2 a3,3 ]
    		B = 
       [ 1 0 0 ]
   [ 0 0 1 ]
   [ 0 1 0 ]
    verify the following:
    1. det(A) = det(AT)
    2. det(AB) = det(A)det(B)

  7. For square matrices in general, does det(AB) = det(BA)? Explain.