Find the cofactor of a matrix entry

Delete the entry’s row and column. Find the determinant of the remaining matrix, then apply the sign from the entry’s position.

Work through a 3×3 example

Use the matrix with rows (3, 2, 8), (1, 7, 4), and (5, 3, 1). To find the cofactor of the entry in row 2, column 3, cross out row 2 and column 3.

The remaining rows are (3, 2) and (5, 3). Their determinant is 3×3 − 2×5 = −1. Row 2 plus column 3 is odd, so the sign is negative. The cofactor is therefore +1.

Use one whole row to find the determinant

The cofactors along row 2 are 22, −37, and 1. Multiply each by its matrix entry and add the three products. The determinant is 1×22 + 7×(−37) + 4×1 = −233. Expanding another row or a column gives the same determinant.

Build the adjugate

Calculate the cofactor for each entry. The adjugate is the transpose of that cofactor matrix. Cofactors also work with fractions. Keep exact values through the expansion to avoid introducing rounding early.

For a derivation and more examples, see MIT OpenCourseWare’s lecture on determinant formulas and cofactors.