In many families of distributions, maximum likelihood estimation is intractable because the normalization constant for the density which enters into the likelihood function is not easily available. The score matching estimator Hyvärinen (1905) provides an alternative where this normalization constant is not required. For an exponential family, e.g. a Gaussian linear concentration model, the corresponding estimating equations become linear (Almeida and Gidas 1993) and the score matching estimator is shown to be consistent and asymptotically normally distributed as the number of observations increase to infinity, although not necessarily efficient. For linear concentration models that are also linear in the covariance (Jensen 1988) we show that the score matching estimator is identical to the maximum likelihood estimator, hence in such cases it is also efficient. Gaussian graphical models and graphical models with symmetries (Højsgaard and Lauritzen 2008) form particularly interesting subclasses of linear concentration models and we investigate the potential use of the score matching estimator for this case.