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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)cofactor(m,i,j):=(-1)^(i+j)*determinant(minor(m,i,j));
                                    i + j
(%o1)     cofactor(m, i, j) := (- 1)      determinant(minor(m, i, j))
(%i2) m1:matrix([1,k_12,k_13],[k_12,1,k_23],[k_13,k_23,1]);
                             [  1    k_12  k_13 ]
                             [                  ]
(%o2)                        [ k_12   1    k_23 ]
                             [                  ]
                             [ k_13  k_23   1   ]
(%i3) determinant(m1);
              2
(%o3)   - k_23  - k_12 (k_12 - k_13 k_23) + k_13 (k_12 k_23 - k_13) + 1
(%i4) ratsimp(determinant(m1)/(-cofactor(m1,1,2)));
                     2                          2       2
                 k_23  - 2 k_12 k_13 k_23 + k_13  + k_12  - 1
(%o4)            --------------------------------------------
                               k_13 k_23 - k_12
(%i5) ratsimp(determinant(m1)/(cofactor(m1,1,1)+cofactor(m1,1,2)+cofactor(m1,1,3)));
                     2                          2       2
                 k_23  - 2 k_12 k_13 k_23 + k_13  + k_12  - 1
(%o5)           ----------------------------------------------
                    2
                k_23  + (- k_13 - k_12) k_23 + k_13 + k_12 - 1
(%i6) cofactor(m1,1,1)+cofactor(m1,1,2)+cofactor(m1,1,3);
                     2
(%o6)          - k_23  + k_13 k_23 + k_12 k_23 - k_13 - k_12 + 1
(%i7) 
Run Example
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) 

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