Chapter 4 — Determinants

NCERT Class 12 Maths Chapter 4 covers Determinants — a number associated with every square matrix that determines properties such as uniqueness of solutions to linear equations, with applications in finding areas of triangles and solving systems of equations using the matrix method.

Chapter 4 of NCERT Class 12 Mathematics Part I introduces determinants of square matrices up to order three with real entries. The chapter covers computation of determinants by expansion along rows or columns, properties of minors and cofactors, the adjoint and inverse of a matrix, and the condition for a matrix to be singular or non-singular. Key applications include finding the area of a triangle using determinants, checking consistency of a system of linear equations, and solving two- or three-variable linear systems using the matrix method (X = A⁻¹B when |A| ≠ 0).

1. 01The determinant of a 2×2 matrix A = [[a11,a12],[a21,a22]] is defined as a11·a22 − a21·a12; only square matrices have determinants.
2. 02A 3×3 determinant can be expanded along any row or column — all six expansions yield the same value; expanding along the row or column with the most zeros simplifies calculation.
3. 03The minor Mij of element aij is the determinant obtained by deleting the ith row and jth column; the cofactor is Aij = (−1)^(i+j) · Mij.
4. 04The adjoint of matrix A is the transpose of the cofactor matrix; A⁻¹ = (1/|A|) · adj A, which exists if and only if |A| ≠ 0 (non-singular matrix).
5. 05Area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) equals (1/2)|det([x1,y1,1; x2,y2,1; x3,y3,1])|; three points are collinear when this determinant is zero.
6. 06For the system AX = B: a unique solution X = A⁻¹B exists when |A| ≠ 0; if |A| = 0 and (adj A)B ≠ O the system is inconsistent; if |A| = 0 and (adj A)B = O it may or may not be consistent.