Determinant of a sub-matrix of the classical adjoint

For me, a standard book to look for such things is Prasolov's linear algebra. This is Theorem 1.2.6.1. Here is the Russian edition, but the proof is essentially a formula.

You might try using the very interesting combinatorial approach of Doron Zeilberger in A Combinatorial Approach to Matrix Algebra, Discrete Mathematics 56 (1985), 61-72. There he gives short proofs of various matrix results such as the Cayley-Hamilton Theorem and the Matrix-Tree Theorem as combinatorial identities, which for me is the "right" way to understand these, rather than appealing to, for example, properties of complex matrices and eigenvalues.