Hint For The Proof: Obvious

Nothing strikes fear into a math student's heart quite like the word "obvious" in a textbook. You're staring at some property about determinants, desperately trying to understand why det(cA) = c^n det(A), and the textbook just casually drops "PROOF. Obvious." with a little checkbox like you're supposed to nod knowingly and move on.

Spoiler alert: it's never obvious. What's obvious to a mathematician who's been doing linear algebra for 30 years is NOT obvious to someone who just learned what a determinant is last week. That little "Obvious" is doing some seriously heavy lifting—probably hiding about 47 steps of algebraic manipulation and three lemmas you haven't covered yet.

Textbook authors really said "I'm not gonna show my work" and expected everyone to be cool with it. The academic equivalent of "trust me bro." Meanwhile, you're spending 2 hours trying to figure out why multiplying one row by a constant only multiplies the determinant by that constant once, not n times. Fun fact: that's actually the difference between Properties 4 and 5—one scales the entire matrix, the other just one row!