3x3 Matrix Determinant — Cofactor Expansion Method

3x3 Determinant via Cofactor Expansion

For a 3x3 matrix:

A = | a  b  c |
    | d  e  f |
    | g  h  i |

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

This is cofactor expansion along the first row. Each term multiplies an element by its cofactor — the determinant of the 2x2 minor matrix obtained by deleting that element's row and column, with an alternating sign pattern (+, -, +).

Step-by-Step Example

A = | 1  2  3 |
    | 4  5  6 |
    | 7  8  9 |

Minor M11 = | 5  6 | = 45 - 48 = -3
            | 8  9 |

Minor M12 = | 4  6 | = 36 - 42 = -6
            | 7  9 |

Minor M13 = | 4  5 | = 32 - 35 = -3
            | 7  8 |

det(A) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0

The determinant is 0, which means this matrix is singular — its rows are linearly dependent (row 3 = 2*row 2 - row 1).

Sarrus' Rule (Alternative)

For 3x3 only, you can also use Sarrus' rule: copy the first two columns to the right, then sum the products along the three downward diagonals and subtract the products along the three upward diagonals.

Computational Note

Cofactor expansion runs in O(n!) time, making it impractical for large matrices. For matrices larger than 4x4, LU decomposition or Gaussian elimination are preferred, reducing the complexity to O(n^3).