Minors and Cofactors Explained: Compute Determinants Easily

The minor M_ij of an entry a_ij is the determinant of the smaller matrix left after deleting row i and column j. The cofactor C_ij is the signed minor: C_ij = (−1)^i+j·M_ij, so it equals +M_ij when i + j is even and −M_ij when i + j is odd.

Minors and cofactors matter because they are the machinery inside every determinant expansion and every matrix inverse. The cofactor expansion formula det(A) = Σ a_ijC_ij along any row or column is how 3 × 3 (and larger) determinants are actually computed, and the adjoint matrix — the transpose of the cofactor matrix — gives the inverse via A^−1 = adj(A)/det(A).

Key Concepts

• Minor M_ij: delete row i and column j, then take the determinant of what remains. For a 3 × 3 matrix each minor is a 2 × 2 determinant.
• Cofactor C_ij: (−1)^i+j·M_ij — the minor with a checkerboard sign attached.
• Sign checkerboard: for 3 × 3 the signs are [[+, −, +], [−, +, −], [+, −, +]], starting with + at (1,1).
• Cofactor expansion: det(A) equals the sum of (entry × its cofactor) along ANY single row or any single column.

Formulas

• Minor: M_ij = det(matrix with row i and column j removed).
• Cofactor: C_ij = (−1)^i+jM_ij.
• Expansion along row i: det(A) = a_i1C_i1 + a_i2C_i2 + … + a_inC_in. Same with a column.
• Inverse: A^−1 = (1/det A)·adj(A); for 2 × 2 this reproduces [[d, −b], [−c, a]]/(ad − bc).

Worked Example

Find all cofactors of row 2 and the determinant of A = [[3, 0, 2], [1, 4, −1], [0, 5, 2]].

1. M_21: delete row 2, column 1 → [[0, 2], [5, 2]]; det = 0·2 − 2·5 = −10. Sign (−1)^3 = −, so C_21 = +10.
2. M_22: delete row 2, column 2 → [[3, 2], [0, 2]]; det = 3·2 − 2·0 = 6. Sign (−1)^4 = +, so C_22 = 6.
3. M_23: delete row 2, column 3 → [[3, 0], [0, 5]]; det = 3·5 − 0·0 = 15. Sign (−1)^5 = −, so C_23 = −15.
4. Expand along row 2: det = a_21C_21 + a_22C_22 + a_23C_23 = 1(10) + 4(6) + (−1)(−15) = 10 + 24 + 15 = 49.
5. Verify along column 1: need C_11: minor [[4,−1],[5,2]] → 8+5 = 13, sign + → 13; C_31: minor [[0,2],[4,−1]] → 0−8 = −8, sign (−1)^4 = + → −8. det = 3(13) + 1(10) + 0(−8) = 39 + 10 = 49. Matches.

Common Misconceptions

• "Minor and cofactor are the same thing." They differ by the sign (−1)^i+j — on the − squares of the checkerboard they are negatives of each other.
• "You must expand along the first row." Any row or column works and gives the same determinant; picking the one with the most zeros is the professional move.
• "The adjoint is just the cofactor matrix." The adjoint is the TRANSPOSE of the cofactor matrix — forgetting the transpose gives a wrong inverse.

Real-World Applications

• Matrix inversion: the adjoint formula A^−1 = adj(A)/det(A) is how inverses are derived theoretically and computed for small matrices in graphics and robotics.
• Cramer's rule: solving AX = B expresses each unknown as a determinant ratio — each numerator is a cofactor-style expansion.
• Computer graphics: inverting 3 × 3 and 4 × 4 transformation matrices (rotations, camera moves) runs on cofactor computations.

A minor is the determinant after deleting one row and column; a cofactor adds the checkerboard sign. Expanding as Σ a_ij·C_ij computes the determinant, and the transposed cofactor matrix (the adjoint) divided by the determinant yields the inverse.