An m \times \ n matrix over a polynomial ring is unimodular if its maximal minors generate the entire ring. If m \leq \ n then this property is equivalent to the matrix being right-invertible and if m \geq \ n then this property is equivalent to the matrix being left-invertible.
i1 : R = QQ[x,y,z]
o1 = R
o1 : PolynomialRing
|
i2 : A = matrix{{x^2*y+1,x+y-2,2*x*y}}
o2 = | x2y+1 x+y-2 2xy |
1 3
o2 : Matrix R <-- R
|
i3 : isUnimodular A
o3 = true
|
i4 : B = matrix{{x*y+x*z+y*z-1},{x^2+y^2}, {y^2+z^2}, {z^2}}
o4 = | xy+xz+yz-1 |
| x2+y2 |
| y2+z2 |
| z2 |
4 1
o4 : Matrix R <-- R
|
i5 : isUnimodular B
o5 = true
|
i6 : I = ideal(x^2,x*y,z^2)
2 2
o6 = ideal (x , x*y, z )
o6 : Ideal of R
|
i7 : isUnimodular presentation module I
o7 = false
|