Cue Column:

• What happens when you swap rows?
• What is det(kA) vs k*det(A)?
• How does transpose affect det?

Note Column: Core properties that save computation time:

1. det($A^T$) = det(A) -- row and column operations are interchangeable
2. Swapping two rows/columns flips the sign
3. Two identical rows/columns => det = 0
4. Multiplying one row by k => det multiplied by k
5. det(kA) = $k^n$ * det(A) for n x n matrix (THE classic trap)
6. det(AB) = det(A) * det(B)
7. Adding a multiple of one row to another does NOT change det
8. If a row is all zeros, det = 0
9. det(A^(-1)) = 1/det(A)
10. For triangular matrices, det = product of diagonal entries

Summary: Properties 5 and 7 are the most tested. Property 5 catches students every year. Property 7 is the workhorse for simplification.