Revision history [back]

This is typical---it is assumed that you can divide and things aren't zero. This happens in other math software systems too. The Maple chapter of the Handbook of Linear Algebra discusses this (p. 72-10) and refers to these two references:

• R.M. Corless and D.J. Jeffrey. Well... it isn’t quite that simple. SIGSAM Bull., 26(3):2–6, 1992.
• R.M. Corless and D.J. Jeffrey. The Turing factorization of a rectangular matrix. SIGSAM Bull., 31(3):20–28, 1997.

That HLA chapter suggests using LU decomposition to do this, but apparently LU decomposition doesn't work too well for our symbolic matrix.

HOWEVER: You can do this in Sage if you use a different base ring for the matrices:

sage: R.<b1,b2,b3>=QQ[]
sage: A=matrix([[1,1,2,b1],[1,0,1,b2],[2,1,3,b3]])
sage: A.echelon_form()
[            1             0             1            b2]
[            0             1             1       b1 - b2]
[            0             0             0 -b1 - b2 + b3]

Note that rref still gives the answer you got before since rref works over the fraction field:

sage: A.rref()
[1 0 1 0]
[0 1 1 0]
[0 0 0 1]
sage: parent(A)
Full MatrixSpace of 3 by 4 dense matrices over Multivariate Polynomial Ring in b1, b2, b3 over Rational Field
sage: parent(A.echelon_form())
Full MatrixSpace of 3 by 4 dense matrices over Multivariate Polynomial Ring in b1, b2, b3 over Rational Field
sage: parent(A.rref())
Full MatrixSpace of 3 by 4 dense matrices over Fraction Field of Multivariate Polynomial Ring in b1, b2, b3 over Rational Field