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In this paper, we study an analytic curve $\varphi: I=[a,b]\rightarrow\mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices,and show that if $\varphi$ satisfies certain geometric condition, then foralmost every point on the curve, the Diophantine approximation given byDirichlet's Theorem can not be improved. To do this, we embed the curve intosome homogeneous space $G/\Gamma$, and prove that under the action of someexpanding diagonal subgroup $A= \{a(t): t \in \mathbb{R}\}$, the translates ofthe curve tend to be equidistributed in $G/\Gamma$, as $t \rightarrow +\infty$.

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices