Determinants

Determinants introduces a single numerical value that can be calculated from any square matrix. The chapter begins with the basic computation of determinants for 2x2 and 3x3 matrices, using the method of expansion along a row or column. This involves understanding the concepts of minors and cofactors, which are the building blocks for all determinant calculations.

The main focus then shifts to the properties of determinants. These are a set of rules that describe how the value of a determinant changes under various operations, such as interchanging rows, multiplying a row by a scalar, or adding a multiple of one row to another. Mastering these properties is essential, as they allow for the simplification of complex determinants without full expansion. This is often the most challenging part of the chapter for students.

Finally, the chapter covers the applications of determinants in geometry and algebra. You will learn to find the area of a triangle given its vertices, test for the collinearity of three points, and solve systems of linear equations using Cramer's Rule. The concept is also used to find the inverse of a matrix through its adjoint. This chapter relies heavily on a prior understanding of Matrices and their basic operations.

Formulas for Determinants

Determinant of a 2x2 matrix: $|A| = ad - bc$ For a matrix A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.
Cofactor of an element: $C_{ij} = (-1)^{i+j} M_{ij}$ M_{ij} is the minor of the element a_{ij}.
Determinant by cofactor expansion: $|A| = \sum_{j=1}^{n} a_{ij} C_{ij}$ Expansion along the i-th row. The result is the same for any row or column.
Area of a triangle: $\text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$ For a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃).
Determinant of a product of matrices: $|AB| = |A| |B|$ For square matrices A and B of the same order.
Determinant of a scalar multiple: $|kA| = k^n |A|$ A is an n x n matrix and k is a scalar.
Property of the Adjoint: $A(\text{adj}(A)) = (\text{adj}(A))A = |A|I$ I is the identity matrix of the same order as A.
Inverse of a matrix: $A^{-1} = \frac{1}{|A|} \text{adj}(A)$ The inverse exists only if |A| ≠ 0.
Determinant of the Adjoint: $|\text{adj}(A)| = |A|^{n-1}$ A is a square matrix of order n.
Cramer's Rule: $x = \frac{\Delta_x}{\Delta}, y = \frac{\Delta_y}{\Delta}, z = \frac{\Delta_z}{\Delta}$ For a system of linear equations, provided the coefficient determinant Δ ≠ 0.

One of these questions, worked through

JEE Advanced 2016

Let $a, \lambda, \mu \in \mathbb{R}$ . Consider the system of linear equations $ax + 2y = \lambda$ $3x - 2y = \mu$ Which of the following statement(s) is (are) correct?

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$ax+2y=\lambda$

$3x-2y=\mu$

$a,\ \lambda,\ \mu \in \mathbb{R}$

$\text{To find: which statements about consistency are correct}$

A= a 2 3 -2 , D= a 2 3 -2

$D=a(-2)-(2)(3)=-2a-6=-2(a+3)$

$\text{Plan: } D\neq 0 \Rightarrow \text{unique};\quad D=0 \Rightarrow \text{check consistency}$

$a\neq -3:\ D\neq 0 \Rightarrow \text{unique solution for all } \lambda,\mu \;\Rightarrow\; (2)\ \text{TRUE}$

$a=-3:\ -3x+2y=\lambda,\qquad 3x-2y=\mu$

$3x-2y=\mu \;\xrightarrow{\times(-1)}\; -3x+2y=-\mu$

$\text{Consistent} \iff \lambda=-\mu \iff \lambda+\mu=0$

$\lambda+\mu=0 \Rightarrow \text{coincident lines} \Rightarrow \infty \text{ solutions} \;\Rightarrow\; (3)\ \text{TRUE}$

$\lambda+\mu\neq 0 \Rightarrow \text{parallel lines} \Rightarrow \text{no solution} \;\Rightarrow\; (4)\ \text{TRUE}$

$(1)\ \text{false: } a=-3 \text{ gives } \infty \text{ only if } \lambda+\mu=0$

$\text{Correct: } (2),\ (3),\ (4)$

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Past-paper questions on Determinants

JEE Advanced 33 questions