The 3×3 Determinant

Linear Algebra (Matrices) · Axiom Academy

Cofactor expansion: break a 3×3 into signed 2×2 pieces by deleting a row and a column, then add them up. 1. The Minor: Delete a Row and a Column Every entry a_ ij has a minor M_ ij : cross out the entry's row i and column j , and the four numbers left over form a 2×2 matrix. Its determinant is the minor. Watch it for a_ 11 =2 — row 1 and column 1 vanish, and the surviving lifts out. Delete row 1 and column 1 → the minor is the 2×2 that remains 2. The Sign: the Checkerboard Cofactor A minor becomes a cofactor once you attach a sign. The sign at position (i,j) is (-1)^ i+j , which lays out as a fixed checkerboard — + in the top-left corner, then alternating. The animation fills it in, then promotes the minor M_ 11 to the cofactor C_ 11 by gluing on its + . here the sign is + , so C_ 11 =+18 3. Cofactor Expansion: Sweep the Row, Add It Up Now run the machine across an entire row. Expanding along row 1 : for each entry, strike its row and column, take the signed 2×2 minor, and add to a running total. The sweep below accumulates the three terms into . You may expand along any row or column — the answer is the same. A zero entry kills its whole term, so a line with zeros is far less work. Each 3×3 cofactor is a 2×2 determinant. A 4×4 would expand into 3×3 cofactors, and so on down — the same rule at every size. You can now compute any 3×3 determinant by cofactor expansion — and the same recipe scales to every larger matrix.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.