Here's a link:

Determinant with Row Multiplied by Constant - ProofWiki
This page shows all but one step of the proof you want, in formal language.

In general, proofwiki is a good place to look for proofs.

BTW, the idea of a determinant as a cofactor expansion was mysterious to me for a long time. I could do the expansions, but I didn't feel I understood why it worked, though I was shy about saying so. I've found that many other people are in the same boat. Suggestion: Google "Geometric Algebra Primer" by Jaap Suter, and/or watch the first few eps of Norman Wildberger's "Wild Lin Alg," which is on Youtube.