Answer to Quiz 6 Fri 18 Feb 2005

1. Consider the matrix

             [  6.0E+9   -5.0E+9  ]
         A = [                    ]
             [ -5.0E+0    4.0E+0  ]
   

   with inverse

        -1   [ -4.0E-9  -5.0E+0  ]
       A   = [                   ]
             [ -5.0E=9  -6.0E+0  ]
   

   Use ||.||_1 to compute the condition number of A.

   Answer:

       ||A||_1 = 6.0E+9 + |-5.0E+0| = 6.0E+9
   
       ||A^(-1)||_1 = |-5.0E+0| + |-6.0E+0| = 1.1E+1
   
       cond(A) = ||A||_1 ||A^(-1)||_1 = 6.6E+10
   

2. Find the LU decomposition of

             [  2  -2   1  ]
         A = [  4  -3   4  ]
             [ -2   3   4  ]
   

   using row pivoting.

   Answer:

                    [  4  -3   4  ] 2
     A -----------> [  2  -2   1  ] 1
                    [ -2   3   4  ] 3
   
               2
   R2 := R2 - --- R1   [    4    -3     4   ] 2
               4       [                    ]
   ------------------> [ (1/2)  -1/2   -1   ] 1
              -2       [                    ]
   R3 := R3 - --- R1   [(-1/2)   3/2    6   ] 3
               4
               2
                       [    4    -3     4   ] 2
                       [                    ]
   ------------------> [(-1/2)   3/2    6   ] 3
                       [                    ]
                       [ (1/2)  -1/2   -1   ] 1
   
              -1/2
   R3 := R3 - ---- R2  [    4    -3     4   ] 2
               3/2     [                    ]
   ------------------> [(-1/2)   3/2    6   ] 3
                       [                    ]
                       [ (1/2) (-1/3)   2   ] 1
   
       [   1     0    0  ]       [  4   -3   4  ]
   L = [ -1/2    1    0  ]   U = [  0   3/2  6  ]
       [  1/2  -1/3   1  ]       [  0    0   2  ]
   
       [ 0 1 0 ]
   P = [ 0 0 1 ]
       [ 1 0 0 ]
   
   L*U = P*A