# Expressing determinant as a linear combination of minors of fixed dimension

Suppose $k<n$. How does one express $\det\begin{pmatrix}a_1^1&\dots&a_n^1\\ \vdots&\ddots&\vdots\\ a^n_1&\dots&a^n_n\end{pmatrix}$ in terms of a linear combination of determinants $\det\begin{pmatrix} a_1^{i_1}&\dots&a_k^{i_1}\\ \vdots&\ddots&\vdots\\ a^{i_k}_1&\dots&a^{i_k}_k\end{pmatrix},$ where $1\leq i_1<i_2<i_3<\cdots<i_k\leq n$?

Just develop according to the last column $(n-k)$ times.