Chapter 4: Determinants

A determinant is a single number that you compute from a square matrix. Despite its modest appearance, it carries enormous information: a matrix is invertible exactly when its determinant is nonzero, the area of a triangle is half the absolute value of a determinant of its vertex coordinates, and a system $AX = B$ has a unique solution iff $\det A \neq 0$ . This chapter develops the algebra of determinants and uses it to invert matrices and solve linear systems via Cramer's rule and the adjoint method.

The historical name "determinant" reflects the fact that the number determines whether a system has a unique solution. In modern language, the determinant equals the signed volume of the parallelepiped spanned by the rows (or columns) of the matrix. A zero determinant means the rows are linearly dependent and the parallelepiped collapses.

Computationally, the chapter is heavy on $2 \times 2$ and $3 \times 3$ determinant expansions. You will learn the cofactor expansion along any row or column, develop a toolkit of seven core properties for simplifying determinants, and use these to evaluate determinants without expanding every entry. JEE problems lean on the properties; board questions are often more direct.

A practical tip: when you see a determinant problem, do not immediately expand. First inspect whether row or column operations simplify the determinant. A clever choice can turn a $3 \times 3$ determinant of unwieldy expressions into a $1 \times 1$ multiple of an easy expression.

The chapter closes with the adjoint method for finding the inverse of a matrix, and Cramer's rule for solving linear systems. These methods are theoretically clean and become indispensable in proofs, even if for hand computation row reduction is often faster.

What's inside

Definition of determinant , $2 \times 2$ and $3 \times 3$ formulas.
Properties of determinants , the seven properties used everywhere.
Area of a triangle , a geometric application.
Minors and cofactors , extracting smaller determinants.
Adjoint and inverse , explicit formula for $A^{-1}$ .
Applications: linear systems , Cramer's rule and the matrix method.

Key results / Formula card

Concept	Formula
$2 \times 2$ determinant	$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$
$3 \times 3$ determinant	$a(ei - fh) - b(di - fg) + c(dh - eg)$
Cofactor $C_{ij}$	$(-1)^{i+j} M_{ij}$ , with $M_{ij}$ = minor
Expansion along any row	$\det A = \sum_j a_{ij} C_{ij}$
Adjoint	$\text{adj}\,A$ = transpose of cofactor matrix
Inverse	$A^{-1} = \tfrac{1}{\det A}\,\text{adj}\,A$
$\det(AB)$	$\det A \cdot \det B$
$\det A^T$	$\det A$
$\det(\lambda A)$	$\lambda^n \det A$ for $n \times n$ matrix
Area of triangle	$\tfrac{1}{2}\,\big\\|\det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix}\big\\|$
Cramer's rule	$x_i = \det A_i / \det A$

How to read this chapter

Start with the definitions. Memorise the seven properties , they are used in nearly every problem. Then practise computing the adjoint of a $3 \times 3$ matrix; that single calculation underpins both inverses and Cramer's rule.

Chapter 4: Determinants

What's inside

Key results / Formula card

How to read this chapter

Sub-topics

Practice quiz