TITLE: Determinants and cofactors?
QUESTION [6 upvotes]: My professor gave us this definition for determinants for a $n \times n$ matrix $A$:
$$\det(A) = a_{11}C_{11} + a_{12}C_{12} ... + a_{1n}C_{1n} $$
where $C_{1j}$ is the cofactor of $A$ on $a_{ij}$. He also said that this can be generalized to:
$$\det(A) = a_{q1}C_{q1} + a_{q2}C_{q2} ... + a_{qn}C_{qn} $$
for any row or column $q$. He didn't give us a proof of the above statement, because he said it was too complicated.
Out of curiosity, could someone give me a proof of the above statement? I would appreciate a proof instead of a tip, because of me being completely new to this and the proof being way too advanced for my skills.
What I know so far: Row Reduction, Basis (not change of basis however), Linear Independence, Span, Transformations, Inverses, Subspaces and Dimensions. Please try to keep proofs within my what I know.
Thank you.

REPLY [2 votes]: This article answers your question exactly. It is 9 pages long and, even then, the actual proof of the fact that the determinant can be calculated along any row or column is sketched.