Math 54(R) Homework #2 Answer Key
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 =============================  Hill section 1.2  =============================

17.  (-2, 1, 3, -1)
     Note: elimination yields the REF system
     [2  3 -1  1 | -5 ]
     [0 -1  4 -3 | 14 ]
     [0  0  6 -6 | 24 ]
     [0  0  0 -5 |  5 ]
     

19.  (-1/2 t + 3/2 s + 3/2, s, -3 t + 2, -3 t + 5, t),
         where s and t are arbitrary.
     Note: elimination yields the REF system
     [2 -3  1 -1  1 |  0]
     [0  0 -1  1  0 |  3]
     [0  0  0 -1 -3 | -5]


21.  None: system is inconsistent.
     Note: elimination yields the REF system
     [3 -1  1 -4 |  2]
     [0  5 -3  4 | -1]
     [0  0  0  0 |  1]
     (More directly: adding the first two equations yields one that
     contradicts the third.)

 =============================  Hill section 1.3  =============================

 2.
         [1 2 3]      [1/2 1/3 1/4]
     A = [2 4 6]  B = [1/3 1/4 1/5]

4.
             [1 -1  0]       [8  2  6]       [ 3 2 3]
     A + B = [2 -1 -2]  2A = [6 -2 -4]  -B = [ 1 0 0]
             [8 -2  0]       [0  0  8]       [-8 2 4]

5.
             [10 0 -1 -3]       [14 4   0 -6]       [-3  2 1 0]
     A + B = [ 3 4 -5 -6]  2A = [-4 0 -10  4]  -B = [-5 -4 0 8]

7.
          [ 0  1]       [-13 -20]
     AB = [-4 -1]  BA = [  8  12]

8.
          [-4 4  -7]       [-7 0  8]
     AB = [11 8 -14]  BA = [ 1 2 12]
          [12 0  -3]       [-8 2  6]

9.
          [6 -10]       [9 2 -9]
     AB = [0   1]  BA = [0 1  0]
                        [3 2 -3]

13. AB and BA are both undefined.

15.
                  [-10  5]            [-8  6]
     (A-B)(A+B) = [ -2 -7]  A^2-B^2 = [-2 -9]

21. Both methods should yield the same answer:
    [-7 -11][s]   [2]
    [36   8][t] = [4]

33.  [Matrix equation should be verified by some sort of hard labor.]
     [-6] =    [9]     [ 4]
     [-6] = -2 [0] + 3 [-2]
     [ 0] =    [3]     [ 2]