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I'm not sure why this does not work over the symbolic ring, but it does so over the rational field of polynomials in h and k:

R.<h,k> = PolynomialRing(QQ,2)
A = matrix([[1,-3,0,2],[-2,4,0,h],[0,-2,1,k],[3,-1,7,1]])
A.echelon_form()

gives

[              1               0               0      -3/2*h - 4]
[              0               1               0      -1/2*h - 2]
[              0               0               1      -h + k - 4]
[              0               0               0 11*h - 7*k + 39]

I'm not sure why this does not work over the symbolic ring, but it does so over the rational field of polynomials in h and k:

R.<h,k> = PolynomialRing(QQ,2)
A = matrix([[1,-3,0,2],[-2,4,0,h],[0,-2,1,k],[3,-1,7,1]])
A.echelon_form()

gives

[              1               0               0      -3/2*h - 4]
[              0               1               0      -1/2*h - 2]
[              0               0               1      -h + k - 4]
[              0               0               0 11*h - 7*k + 39]

ADDED. Explanation for why this does not work over SR is given in this answer.