YOU must solve these via matrices.
It's ridiculous to try to solve it via the normal equations.
My most recent class listened and did very well on the exam.
The model is
Where is a column vector or your N(0,sigma^2) errors.
The Y is a column vector of your data.
is a column vector of your parameters.
The LSE of is .
X is the design matrix. The columns are the data via , x , x squared.... depending on the model.
and MSE, your unbiased estimator of sigma^2 is SSE/(n-number of parameters in the beta vector).
page 8,9 of http://malroy.econ.ox.ac.uk/fisher/iss/isslecture1.pdf is decent