Exercise 6, § 8

Show that, if $X$ is a matrix with $n$ rows and $m$ columns, $Y$ a matrix with $p$ rows and $n$ columns and $Z = YX$ the product matrix with $p$ rows and $m$ columns, the minors of $Z$ of order $q$ are zero if $n < q$; if $q \leq n$, they are given by

where K runs through the set of subsets of $\{1, n\}$ with $q$ elements (use formula (3) of § 7, no. 2).