Determinants

The essentials

Every square matrix A has a number called its determinant, written |A| or det A. For a 2 × 2 matrix [[a, b], [c, d]], |A| = ad − bc. A 3 × 3 determinant is expanded along any row or column using cofactors.

The minor Mᵢⱼ of element aᵢⱼ is the determinant left after deleting row i and column j. The cofactor is Aᵢⱼ = (−1)^(i+j) Mᵢⱼ. The determinant equals the sum of the products of the elements of any one row (or column) with their own cofactors. If elements of one row are multiplied by cofactors of a different row, the sum is 0.

Key points

• Interchanging two rows (or columns) changes the sign of the determinant.
• If two rows or columns are identical or proportional, the determinant is 0.
• Multiplying one row by k multiplies the determinant by k; for an n × n matrix, |kA| = kⁿ|A|.
• Adding a multiple of one row to another leaves the determinant unchanged.
• |Aᵀ| = |A| and |AB| = |A||B|.
• Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½ |det[[x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1]]|.
• Three points are collinear exactly when this determinant is 0.

Exam tips

• Expand along the row or column with the most zeros to save time.
• Area is always positive, so take the absolute value at the end.

Common mistakes

• Forgetting the sign pattern + − + for cofactors in the first row.
• Writing |kA| = k|A| for a 3 × 3 matrix instead of k³|A|.