01 Determinants
The 2×2 determinant
- For A = [[a, b], [c, d]], det A = |A| = ad − bc.
- Example: det [[3, 5], [2, 4]] = (3)(4) − (5)(2) = 12 − 10 = 2.
- The determinant is a number, never a matrix.
The 3×3 determinant
- Expand along any row or column using cofactors with alternating signs (+ − +).
- Expanding along the first row: |A| = a(ei − fh) − b(di − fg) + c(dh − eg).
- Example: A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]] → 1(0 − 24) − 2(0 − 20) + 3(0 − 5) = −24 + 40 − 15 = 1.
- Practical tip: expand along the row or column containing the most zeros — it shortens the work without changing the answer.
Properties of determinants
1Zero row or column
Statement
- If a whole row or column is zero, the determinant is 0
2Identical rows
Statement
- Two identical rows (or columns) → determinant 0
3Row swap
Statement
- Swapping two rows changes the sign of the determinant
4Scalar multiple
Statement
- Multiplying one row by k multiplies the determinant by k
5Product rule
Statement
- |AB| = |A|·|B|
6Transpose
Statement
- |Aᵀ| = |A|
7Triangular matrix
Statement
- Determinant equals the product of the diagonal entries