By means of a special LU factorization of the Mina matrix with the $n$-th row and $k$-th column entry $\mathbf{D}^{n}_x(f^{a_k}(x))$, we obtain not only a short proof of the Mina determinant identity but also the inverse of the Mina matrix. Finally, by use of some similar factorizations built on the Lagrange interpolation formula, two new determinant identities of Mina type are established.
