Why Properties Matter
Computing a 3×3 determinant from scratch involves many multiplications. Properties of determinants allow us to simplify the matrix first (introduce zeros, reduce rows) and then evaluate it easily. In exams, these properties save enormous time.
The value of a determinant does not change when valid row/column operations are applied (with exceptions noted). The goal is always to make the matrix triangular or introduce zeros.
Property 1 — Transpose
The value of the determinant remains unchanged when rows and columns are interchanged (i.e., when the matrix is transposed).
Implication: Any property that holds for rows also holds for columns and vice versa. This doubles the number of useful operations available to us.
|A| = 1×4 − 2×3 = 4 − 6 = −2
Aᵀ = [[1,3],[2,4]], |Aᵀ| = 1×4 − 3×2 = 4 − 6 = −2 ✓
Property 2 — Row/Column of Zeros
A determinant is zero if all elements of any one row or one column are zero.
Row 2 is all zeros → |A| = 0 (no calculation needed!)
Property 3 — Row Interchange
If any two rows (or columns) of a determinant are interchanged, then the sign of the determinant changes but its magnitude stays the same.
Each swap of rows/columns negates the determinant. Two swaps bring it back to the original. Keep track of the number of swaps!
|A| = 1×4 − 2×3 = −2
After R1↔R2: B = [[3,4],[1,2]], |B| = 3×2 − 4×1 = 6 − 4 = +2 = −(−2) ✓
Property 4 — Identical Rows
When any two rows (or columns) of a determinant are identical (or proportional), the value of the determinant is zero.
Proof sketch: Swap the two identical rows → sign changes → |A| = −|A| → 2|A| = 0 → |A| = 0. Elegant!
This also applies if two rows are proportional (one is a scalar multiple of the other).
R3 = 2×R1. Since rows are proportional → |A| = 0
Property 5 — Scalar Multiple
Multiplying all elements of any one row (or column) by a scalar k multiplies the determinant by k.
Important contrast with matrices: For an n×n matrix, |kA| = kⁿ|A| (scalar applies to ALL rows). But this property is about multiplying just ONE row.
Property 6 — Row Operation
Adding to any row (or column) a scalar multiple of another row (or column) leaves the determinant unchanged.
This is the most powerful property — it's the basis of row reduction. We use it to introduce zeros in the matrix and simplify computation.
Apply R₁ → R₁ − R₂ (or other combinations) to introduce zeros
Apply R₂ → R₂ − R₃ etc.
Expand along the row/column with most zeros
Quick Reference Summary
| # | Property | Effect on |A| |
|---|---|---|
| 1 | Transpose | No change |
| 2 | Row/Column of zeros | |A| = 0 |
| 3 | Swap two rows/columns | Sign changes (×−1) |
| 4 | Identical or proportional rows | |A| = 0 |
| 5 | Multiply one row by k | Multiplied by k |
| 6 | Rᵢ → Rᵢ + k·Rⱼ | No change ⭐ |