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If you want to augment M with symbolic elements it need to be defined over the symbolic ring SR. You can do so upfront via

M=matrix(SR, [[-3,1,1,1],[1,-3,1,1],[1,1,-3,1],[1,1,1,-3]])

or alternatively you can keep the definition of M intact but change its ring just before the augmenation:

AUG=M.change_ring(SR).augment(v,subdivide=true)

If you want to augment M with symbolic elements it need needs to be defined over the symbolic ring SR. You can do so upfront via

M=matrix(SR, M = matrix(SR, [[-3,1,1,1],[1,-3,1,1],[1,1,-3,1],[1,1,1,-3]])

or alternatively Alternatively, you can keep the original definition of M intact intact, but change its ring just before the augmenation:

AUG=M.change_ring(SR).augment(v,subdivide=true)
AUG = M.change_ring(SR).augment(v,subdivide=true)