Determinant (3×3)

This formula computes the determinant of a 3×3 matrix by breaking it into three 2×2 determinants (called minors) with alternating signs. It works because the determinant measures signed volume scaling, and cofactor expansion preserves that value while reducing the problem to smaller matrices. The pattern +, −, + comes from the sign structure of cofactors, which ensures the result is consistent with the determinant’s antisymmetric properties.

Use this when you need the determinant of a 3×3 matrix by hand, especially in exams or when checking whether a matrix is invertible. It is also useful in systems of linear equations, geometry, and vector calculations such as finding volume or verifying coplanarity.

• Forgetting the alternating signs in the expansion and adding all three terms
• Mixing up the entries in the 2×2 minors, especially the order of subtraction
• Using the wrong row or column coefficients when applying cofactor expansion