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minor-rank

m: zeromatrix(1,1);

rank(minor(m,1,1));

Calculate

minor-rank

m: zeromatrix(1,1);

minor(m,1,1);

rank(minor(m,1,1));

Calculate

minor-rank

m: zeromatrix(1,1);

rank(minor(m,1,1));

Calculate

minor-rank

m: zeromatrix(1,1);

minor(m,1,1);

rank(minor(m,1,1));

Calculate

minor

Run Example
(%i1)m: matrix([1,1,0],[1,1,1],[0,1,0]);
                                  [ 1  1  0 ]
                                  [         ]
(%o1)                             [ 1  1  1 ]
                                  [         ]
                                  [ 0  1  0 ]
(%i2) eigenvalues(minor(m,3,3));
(%o2)                          [[0, 2], [1, 1]]
(%i3) r: matrix([c,-s,0],[s,c,0],[0,0,1]);
                                 [ c  - s  0 ]
                                 [           ]
(%o3)                            [ s   c   0 ]
                                 [           ]
                                 [ 0   0   1 ]
(%i4) v: transpose([x,y,1]);
                                     [ x ]
                                     [   ]
(%o4)                                [ y ]
                                     [   ]
                                     [ 1 ]
(%i5) eq: expand(transpose(r.v).m.r.v);
       2  2          2    2  2      2          2                2  2          2
(%o5) s  y  - 2 c s y  + c  y  - 2 s  x y + 2 c  x y + 2 c y + s  x  + 2 c s x
                                                                   2  2
                                                                + c  x  + 2 s x
(%i6) m1: rat(transpose(r).m.r);
                    [  2            2        2    2       ]
                    [ s  + 2 c s + c      - s  + c      s ]
                    [                                     ]
(%o6)/R/            [       2    2      2            2    ]
                    [    - s  + c      s  - 2 c s + c   c ]
                    [                                     ]
                    [        s                c         0 ]
(%i7) l: solve([m1[1,2],m1[2,1],m1[2,2]],[c]);
(%o7)                              [[c = s]]
(%i8) eq1: expand(transpose(v).subst(l[1],m1).v);
                                       2  2
(%o8)                       2 s y + 4 s  x  + 2 s x
(%i9) rat(solve(eq1,y),x);
                                          2
(%o9)/R/                      [y = - 2 s x  - x]
(%i10) 
Run Example
A:matrix([a,b,c],[d,e,f],[g,h,i]);
                                  [ a  b  c ]
                                  [         ]
(%o1)                             [ d  e  f ]
                                  [         ]
                                  [ g  h  i ]
(%i2) A12:minor(A,1,2);
                                   [ d  f ]
(%o2)                              [      ]
                                   [ g  i ]
(%i3) A11:minor(A,1,2);
                                   [ d  f ]
(%o3)                              [      ]
                                   [ g  i ]
(%i4) A22:minor(A,2,2);
                                   [ a  c ]
(%o4)                              [      ]
                                   [ g  i ]
(%i5) A21:minor(A,2,1);
                                   [ b  c ]
(%o5)                              [      ]
                                   [ h  i ]
(%i6) (determinant(A12))^2-determinant(A11)*determinant(A22)+determinant(A12)*determinant(A21);
                  2
(%o6)  (d i - f g)  + (b i - c h) (d i - f g) - (a i - c g) (d i - f g)
(%i7) 
Run Example
pe(x1,y1,z1,x2,y2,z2):=(2*x1*x2)-(2*x1*y2)-(2*y1*x2)+(x1*z2)+(z1*x2)+(3*y1*y2)+(6*z1*z2);
(%o1) pe(x1, y1, z1, x2, y2, z2) := 2 x1 x2 - 2 x1 y2 - 2 y1 x2 + x1 z2 + z1 x2
                                                            + 3 y1 y2 + 6 z1 z2
(%i2) m11:pe(1,0,0,1,0,0);
(%o2)                                  2
(%i3) m12:pe(1,0,0,0,1,0);
(%o3)                                 - 2
(%i4) m13:pe(1,0,0,0,0,1);
(%o4)                                  1
(%i5) m21:pe(0,1,0,1,0,0);
(%o5)                                 - 2
(%i6) m22:pe(0,1,0,0,1,0);
(%o6)                                  3
(%i7) m23:pe(0,1,0,0,0,1);
(%o7)                                  0
(%i8) m31:pe(0,0,1,1,0,0);
(%o8)                                  1
(%i9) m32:pe(0,0,1,0,1,0);
(%o9)                                  0
(%i10) m33:pe(0,0,1,0,0,1);
(%o10)                                 6
(%i11) M:matrix([m11,m12,m13],[m21,m22,m23],[m31,m32,m33]);
                                [  2   - 2  1 ]
                                [             ]
(%o11)                          [ - 2   3   0 ]
                                [             ]
                                [  1    0   6 ]
(%i12) Mt:transpose(M);
                                [  2   - 2  1 ]
                                [             ]
(%o12)                          [ - 2   3   0 ]
                                [             ]
                                [  1    0   6 ]
(%i13) M-Mt;
                                  [ 0  0  0 ]
                                  [         ]
(%o13)                            [ 0  0  0 ]
                                  [         ]
                                  [ 0  0  0 ]
(%i14) determinant(M);
(%o14)                                 9
(%i15) M2:minor(M,2,2);
                                   [ 2  1 ]
(%o15)                             [      ]
                                   [ 1  6 ]
(%i16) determinant(minor(M,2,2));
(%o16)                                11
(%i17) determinant(minor(M,1,1));
(%o17)                                18
(%i18) 

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