Evaluate the determinant with rows [2 1 0], [1 3 −1] and [0 4 2].
- Expand along row 1: 2 × M11 − 1 × M12 + 0 × M13.
- M11 = (3 × 2) − (−1 × 4) = 6 + 4 = 10.
- M12 = (1 × 2) − (−1 × 0) = 2.
- Value = (2 × 10) − (1 × 2) + 0 = 20 − 2 = 18.
Answer: 18
Determinants
Maths · Matrices and determinants. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Every square matrix A has one number attached to it, called its determinant, written |A| or det A. For a 2 × 2 matrix it is the product of the main diagonal minus the product of the other diagonal.
A 3 × 3 determinant is expanded along any one row or column. Each entry is multiplied by its cofactor, and the three products are added. The cofactor is the minor (the 2 × 2 determinant left when the entry's row and column are deleted) with a sign attached.
The properties of determinants often give the value with very little working. A matrix is singular when its determinant is zero.
Rules to remember
- For rows [a b] and [c d]: |A| = ad − bc.
- Cofactor: Aij = (−1)i+j × Mij, where Mij is the minor. The signs follow the pattern + − + / − + − / + − +.
- Expansion along row 1: |A| = a11A11 + a12A12 + a13A13.
- |A| = 0 if a row (or column) is all zeros, or if two rows (or columns) are identical or proportional.
- Interchanging two rows (or columns) changes the sign of the determinant. |At| = |A|.
- Multiplying one row by k multiplies the determinant by k, so for an n × n matrix |kA| = kn|A|.
- Adding a multiple of one row to another row does not change the value. For a triangular matrix, |A| = product of the diagonal entries.
Common mistake
Students forget the minus sign on the middle term when expanding along the first row. The cofactor of a12 is −M12, so the expansion is a11M11 − a12M12 + a13M13.
Worked example
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