Show that for the normal linear model (boldsymbol{Y} sim mathscr{N}left(mathbf{X} beta, sigma^{2} mathbf{I}_{n} ight)), the maximum likelihood estimator of (sigma^{2}) is identical to the method

Show that for the normal linear model \(\boldsymbol{Y} \sim \mathscr{N}\left(\mathbf{X} \beta, \sigma^{2} \mathbf{I}_{n}\right)\), the maximum likelihood estimator of \(\sigma^{2}\) is identical to the method of moments estimator (2.37).