MA 305 Spring 2K Homework 2 Solutions
> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/initlib.mpl"):
> with(refpkg):
> infolevel['refpkg']:=3:
> with(linalg):
Warning, new definition for norm
Warning, new definition for trace
Problem 1
> A := matrix(3,3,[1,2,3,3,1,3,2,1,4]);

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

a.
> evalm(3 * A + 5 * A^2);

                          [68    41    114]
                          [               ]
                          [69    53    129]
                          [               ]
                          [71    48    137]

b.
> I3 := array(identity, 1..3, 1..3);

              I3 := array(identity, 1 .. 3, 1 .. 3, [])

> evalm(A^4 - A^3 + 2 * A^2 - A + 4 * I3);

                         [479    308    861]
                         [                 ]
                         [519    365    975]
                         [                 ]
                         [536    363    998]

Problem 2
> A := matrix(3,2,[1,3,1,4,2,1]);

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

> B := matrix(3,3,[1,1,3,2,1,2,4,2,1]);

                               [1    1    3]
                               [           ]
                          B := [2    1    2]
                               [           ]
                               [4    2    1]

a.
> evalm(transpose(A&*B));
Error, (in linalg[multiply]) non matching dimensions for vector/matrix product
The multiplication is not possible because A; is a 3 x 2 matrix and B; is a 3 x 3 matrix, and the dimensions are not compatible for matrix multiplication.
b.
> evalm(transpose(A)&*transpose(B));

                           [ 8     7     8]
                           [              ]
                           [10    12    21]

c.
> evalm(B&*transpose(B));

                           [11     9     9]
                           [              ]
                           [ 9     9    12]
                           [              ]
                           [ 9    12    21]

d.
> evalm(A&*transpose(A));

                           [10    13    5]
                           [             ]
                           [13    17    6]
                           [             ]
                           [ 5     6    5]

e.
> evalm(transpose(A)&*A);

                              [6     9]
                              [       ]
                              [9    26]

f.
> evalm(transpose(B)&*transpose(A));
Error, (in linalg[multiply]) non matching dimensions for vector/matrix product
The multiplication is not possible because B^T; is a 3 x 3 matrix and A^T; is a 2 x b matrix, and the dimensions are not compatible for matrix multiplication.
Problem 3
> inverse(matrix(4,4,[1,0,0,0,0,3,0,0,2,0,3,0,4,5,0,1]));

                      [ 1       0       0     0]
                      [                        ]
                      [ 0      1/3      0     0]
                      [                        ]
                      [-2/3     0      1/3    0]
                      [                        ]
                      [ -4     -5/3     0     1]

Problem 4
> inverse(matrix(4,4,[1,b,0,0,a,1,b,0,0,a,1,b,0,0,a,1]));

            [                                2        3 ]
            [  -1 + 2 b a   b (-1 + b a)    b        b  ]
            [- ---------- , ------------ , ---- , - ----]
            [      %1            %1         %1       %1 ]
            [                                           ]
            [                                         2 ]
            [a (-1 + b a)      -1 + b a       b      b  ]
            [------------ ,  - -------- ,  - ---- , ----]
            [     %1              %1          %1     %1 ]
            [                                           ]
            [  2                                        ]
            [ a         a        -1 + b a   b (-1 + b a)]
            [---- ,  - ---- ,  - -------- , ------------]
            [ %1        %1          %1           %1     ]
            [                                           ]
            [    3      2                               ]
            [   a      a     a (-1 + b a)     -1 + 2 b a]
            [- ---- , ---- , ------------ , - ----------]
            [   %1     %1         %1              %1    ]

                                            2  2
                         %1 := 1 - 3 b a + b  a

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