Determinants Notes for Class 12 & IIT JEE

A determinant is a unique scalar value associated with every square matrix. For a 2×2 matrix [[a,b],[c,d]], the determinant = ad − bc. For a 3×3 matrix, it is computed by expanding along any row or column using cofactors. Determinants are used to find triangle areas, solve linear systems via Cramer's rule, and check matrix invertibility.

The 10 key properties are: transposition (|A^T| = |A|), row/column swap changes sign, identical rows/columns give zero, scalar multiple, zero row/column gives zero, proportional rows/columns give zero, sum property, row operation leaves value unchanged, triangular matrix determinant equals product of diagonals, and |AB| = |A||B|. Properties 6 and 8 are the most used in JEE simplification.

Area = (1/2)|det M|, where M is the 3×3 matrix with rows [x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1] for the three vertices. The absolute value ensures area is positive. If this determinant equals zero, the three points are collinear — this is the standard collinearity condition tested in JEE Main.

Cramer's rule solves a system of n linear equations in n unknowns using determinants. For AX = B, compute D = det(A), D₁ (D with column 1 replaced by B), D₂, D₃. Then x = D₁/D, y = D₂/D, z = D₃/D — valid only when D ≠ 0. If D = 0, Cramer's rule does not apply; the system is either inconsistent or has infinitely many solutions.

The minor M_ij of element a_ij is the determinant of the submatrix obtained by deleting row i and column j. The cofactor A_ij is the signed minor: A_ij = (−1)^(i+j) × M_ij. Cofactors are used to expand determinants and to compute the adjoint of a matrix, which in turn gives the inverse via A⁻¹ = adj(A)/|A|.