Determinants

part 1

Determinants and Properties

• Determinant of order 1, 2, 3
• Properties of determinants
• Expansion of determinants using cofactors

part 2

Area and Adjoint

• Area of triangle using determinants
• Minors and cofactors
• Adjoint of a matrix
• Inverse of a matrix using adjoint

part 3

System of Linear Equations

• Consistency of system of linear equations
• Solution using Cramer's rule
• Solution using matrix method
• Applications of determinants

2×2 determinant

$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

★ board favourite

Area of triangle (coordinate geometry)

$\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$

★ board favourite

Inverse of matrix

$A^{-1} = \frac{\text{adj}(A)}{|A|},\ |A| \neq 0$

★ board favourite

Adjoint of A

$\text{adj}(A) = [C_{ji}]$ (transpose of cofactor matrix)

★ frequently asked

A·adj(A) = |A|·I

$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| \cdot I$

★ frequently asked

AX = B has unique solution if |A| ≠ 0

$X = A^{-1}B$

★ frequently asked

|AB| = |A|·|B|

$|AB| = |A| \cdot |B|$

★ frequently asked

|kA| = kⁿ|A| for n×n matrix

$|kA| = k^n|A|$

★ frequently asked

Solve the following system of equations using the matrix (inverse) method: x + 2y + z = 4, −x + y + z = 0, x − 3y + z = 4.

Using properties of determinants, prove that |[1, a, a²],[1, b, b²],[1, c, c²]| = (a − b)(b − c)(c − a).

Using elementary row operations, find the inverse of the matrix A = [[1,2,3],[0,2,4],[0,0,5]]. Verify that A·A⁻¹ = I.

Prove that |[a+b, b+c, c+a],[b+c, c+a, a+b],[c+a, a+b, b+c]| = 2|[a,b,c],[b,c,a],[c,a,b]|.

If A is a 3×3 matrix with |A| = 4, find the value of |2A|.

Find the value of k if the area of the triangle with vertices (2, −6), (5, 4) and (k, 4) is 35 square units, using determinants.

If A = [[2, −3, 5],[3, 2, −4],[1, 1, −2]], find A⁻¹ using the adjoint. Hence solve the system: 2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y − 2z = −3.

For what value of x is the matrix A = [[3 − x, 2],[4, 1 − x]] singular?

How many marks is Determinants worth in the Class 12 board exam?

Around 5 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Cofactor expansion and Cramer's rule for 3 equations.

Is Determinants easy or hard?

It's rated hard — one of the tougher chapters in Class 12 Maths. Most students need about 4 hours to cover it well.

What are the important topics in Determinants?

The chapter covers 10 NCERT topics in 3 areas: Determinants and Properties; Area and Adjoint; System of Linear Equations.

What questions come from Determinants in board exams?

Between 2019–2024, CBSE board papers asked questions from this chapter on Solving Equations by Matrix Method, Properties of Determinants, Inverse of Matrix, Value of Determinant — 1- to 5-mark questions.