# determinants of matrices of minors

Let $A$ be an $n \times n$ matrix and fix an integer $k$ with $1 \leq k \leq n$. Define a new matrix $\text{minor}_k(A)$ whose entries are the $k \times k$ minors of $A$. This new matrix will be $\binom{n}{k} \times \binom{n}{k}$.

Theorem? Let $D$ be the determinant of $A$. The determinant of $\text{minor}_k(A)$ is $D^\binom{n-1}{k-1}$.

Is this right? Can anyone provide a reference or proof?

(If you want to be more precise in the definition of $\text{minor}_k(A)$, put an ordering on the cardinality $k$ subsets of $\{1, 2, \dots, n\}$, and index the rows and columns of $\text{minor}_k(A)$ using that ordering. The $(i,j)$ entry is then the determinant of the matrix obtained by keeping only the rows of $A$ indexed by the $i$th subset, and the columns of $A$ indexed by the $j$ subset. Changing the ordering shouldn't affect the determinant of the matrix of minors.)

Note, however, that the theorem requires that the index sequences for producing minors are listed in lexiographic order. Otherwise the determinant may become $-D^\binom{n-1}{k-1}$.