What is a determinant?

Every square matrix has one number attached to it, called its determinant, written |A| or det A. A matrix is a table; a determinant is a single value worked out from the table. (Here |A| has nothing to do with size or distance.)

2 × 2: for [a, b; c, d], |A| = ad – bc: the leading diagonal product minus the other diagonal product. Example: |3, –1; 2, 5| = 15 – (–2) = 17.

3 × 3: break it down into 2 × 2 determinants. Pick a row (usually the first). For each element in it:

• cover its row and column; the 2 × 2 determinant left over is its minor;
• multiply the element by that minor;
• attach a sign from the pattern + – + / – + – / + – +.

Add the three results.

Example: |2, 0, 1; 3, 1, 0; 1, 4, 2| along row 1 = 2(1 × 2 – 0 × 4) – 0(...) + 1(3 × 4 – 1 × 1) = 4 – 0 + 11 = 15.

You may expand along any row or column: the answer is always the same. So pick the one with the most zeros: every zero wipes out one whole term. In the example above, the 0 in row 1 saved one 2 × 2 calculation.

Scaling a matrix. Multiplying an n × n matrix by k multiplies every row by k, and each row brings out its own k. So |kA| = kⁿ|A|. For a 3 × 3 matrix, |5A| = 125|A|, not 5|A|.

Determinants exist only for square matrices: a 2 × 3 table has no determinant.