  In this note we consider some properties of $GL_n(\mathbb{R})$ with the
Semi-Riemannian structure induced by the trace metric $g$. In particular we
study geodesics and curvature tensors. Moreover we prove that $GL_n$ has a
suitable foliation, whose leaves are isometric to $(SL_n(\mathbb{R}), g)$,
while its component of matrices with positive determinant is isometric to the
Semi-Riemannian product manifold $SL_n \times \mathbb{R}$.
