3×3 DeterminantJEE Main

Concept

A 3×3 determinant expands along any row or column: each element times its cofactor (signed minor, chessboard signs + − + / − + − / + − +). Geometrically it's the signed volume of the parallelepiped spanned by the rows — zero means the three vectors are coplanar and the matrix has no inverse.

Key formula

Derivation

The minor $M_{ij}$ deletes row i and column j, leaving a 2×2. Signs follow $(-1)^{i+j}$.

Properties that do the real work: swapping rows flips the sign; a common row factor pulls out; adding a multiple of one row to another CHANGES NOTHING — the royal road to fast evaluation is making zeros first, then expanding along the zero-rich line.

Scenarios to explore

• 3×3 Determinant — Cofactor expansion and the parallelepiped-volume meaning.

Real-world applications

• Cramer's rule for 3-variable systems.
• Scalar triple product & tetrahedron volumes (V = |det|/6).
• Coplanarity/concurrency tests in coordinate geometry.

JEE exam tips

• Make zeros FIRST with row operations, then expand — halves the arithmetic.
• Two identical (or proportional) rows ⇒ det = 0, instantly.
• det(Aᵀ) = det A; det(A⁻¹) = 1/det A; det(adj A) = (det A)² for 3×3.

Common mistakes

• Dropping the alternating signs on cofactors.
• Row operations: R1 → R1 + kR2 is free, but R1 → kR1 multiplies det by k.
• Expanding along a full row when a column has two zeros waiting.

Exam traps to avoid

• det(2A) = 8·detA for 3×3 (2³, not 2).
• System AX = B: unique solution ⟺ det ≠ 0; det = 0 splits into no/infinite solutions via consistency.