Linear complementary dual (LCD) codes are linear codes satisfying $C\cap C^{\perp }=\left\{0\right\}$. Researchers have proved to construct linear codes over finite fields ${\mathbb{F}}_q$, q>3, is equivalent to construct LCD codes. This means that the investigation of binary and ternary LCD codes is the only remaining open problem. In this work, we propose new constructions of binary and ternary LCD codes by using matrix-product codes and prove that binary and ternary LCD codes from matrix-product codes are asymptotically good. Then we construct some new and good binary and ternary LCD codes using matrix-product matrices.