Exercise 9, § 8

Let $A$ be a square matrix of order $n$, $B$ a submatrix of $A$ with $p$ rows and $q$ columns and $C$ the matrix obtained by multiplying in $A$, each of the elements belonging to $B$ by the same scalar $\alpha$. Show that each term in the total expansion of $\det C$ is equal to the corresponding term in the total expansion of $\det A$ multiplied by a scalar of the form $\alpha^r$, where $r \geq p + q - n$ and $r$ depends on the term considered (perform a suitable Laplace expansion of $\det C$).