Continuity of the Inverse of a Continuous Matrix Function

Let $a<b$ be real numbers, let $k\ge1$ be a natural number, and let $M$ assign to each $t\in[a,b]$ an invertible real $k\times k$ matrix $M(t)$ whose entries are continuous functions of $t$ on $[a,b]$, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric.

1. The assignment $t\mapsto M(t)^{-1}$ has entries that are continuous functions of $t$ on $[a,b]$.

2. If in addition every $M(t)$ is symmetric positive definite, then every $M(t)^{-1}$ is symmetric positive definite.