Chapter 4 — Determinants

Chapter 4 of the Class 12 Maths NCERT textbook, "Determinants", covers the 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.

• The determinant as a single test number — Every square matrix condenses to one number, its determinant, computed by expanding along any convenient row or column. This single value signals whether a matrix is singular or non-singular, which governs everything the chapter goes on to do.
• From cofactors to the inverse — The chapter builds minors and cofactors from the matrix, assembles them into the adjoint, and uses it to construct the inverse — showing that a matrix can be inverted exactly when its determinant is non-zero.
• Solving geometry and linear systems — Determinants become a practical tool: they give the area of a triangle from its vertices (and detect collinear points), and they decide whether a system of linear equations has a unique solution, no solution, or infinitely many via the matrix method.

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.