Board exam focus — Determinants (CBSE & HBSE)

Determinants is one of the most heavily tested chapters, carrying 10-12 marks in CBSE. Topics include properties of determinants, cofactor expansion, adjoint, inverse, and solving systems of equations.

Determinants — Evaluation and Properties

Determinant of a Matrix

2×2 Determinant: |A| = |a b; c d| = ad − bc

3×3 Determinant (expansion along first row): |a₁ b₁ c₁; a₂ b₂ c₂; a₃ b₃ c₃| = a₁(b₂c₃−b₃c₂) − b₁(a₂c₃−a₃c₂) + c₁(a₂b₃−a₃b₂)

Properties of Determinants

Area of Triangle

Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃):

Area = (1/2)|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

or equivalently: Area = (1/2) × |det([[x₁,y₁,1],[x₂,y₂,1],[x₃,y₃,1]])|

For collinear points, Area = 0, so the determinant = 0.

CBSE Tip: When evaluating determinants, use properties to create zeros (especially two zeros in a row/column) to simplify expansion. Aim to make at least two elements zero before expanding.

Minors, Cofactors, Adjoint and Inverse

Minors and Cofactors

Minor Mᵢⱼ of element aᵢⱼ: determinant of the matrix obtained by deleting row i and column j.

Cofactor Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ

Sign pattern for cofactors:

+ − +
− + −
+ − +

Expansion using cofactors: |A| = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ (expanding along R₁)

Key result: Expansion along any row or column gives the same value |A|. Cross-expansion: aᵢ₁Cⱼ₁ + aᵢ₂Cⱼ₂ + aᵢ₃Cⱼ₃ = 0 when i ≠ j

Adjoint of a Matrix

adj(A) = transpose of the cofactor matrix (adj A)ᵢⱼ = Cⱼᵢ

Key Property: A·adj(A) = adj(A)·A = |A|·I

Inverse of a Matrix

If |A| ≠ 0, then: A⁻¹ = adj(A) / |A|

Important results:

• |adj A| = |A|^(n-1) for an n×n matrix
• adj(AB) = adj(B)·adj(A)
• (adj A)' = adj(A')

Singular and Non-Singular Matrices

• A is singular iff |A| = 0 (no inverse)
• A is non-singular iff |A| ≠ 0 (inverse exists)

CBSE Tip: For a 3×3 matrix, find ALL nine cofactors systematically (3×3 = 9 minors), then arrange as the cofactor matrix, transpose to get adj(A), and divide by |A|.

Solving Systems of Equations using Matrices

Matrix Method for Solving Linear Equations

System AX = B has:

• Unique solution if |A| ≠ 0: X = A⁻¹B
• No solution or infinitely many solutions if |A| = 0
• If (adj A)·B = O → infinitely many solutions (consistent)
• If (adj A)·B ≠ O → no solution (inconsistent)

Cramer's Rule (for system ax+by=e, cx+dy=f)

For AX = B:

• D = |A| (determinant of coefficient matrix)
• D₁ = |A| with column 1 replaced by B
• D₂ = |A| with column 2 replaced by B
• x = D₁/D, y = D₂/D (if D ≠ 0)

Step-by-Step: Matrix Method

Given: 2x+y+z=3, x−y−z=0, x+2y+3z=9

Step 1: Write A = [[2,1,1],[1,−1,−1],[1,2,3]], X = [[x],[y],[z]], B = [[3],[0],[9]]

Step 2: |A| = 2(−3+2)−1(3+1)+1(2+1) = 2(−1)−4+3 = −3

Step 3: Find A⁻¹ using adjoint or ERO

Step 4: X = A⁻¹B gives unique solution

Consistency Check Summary

CBSE Tip: In 5-mark questions, CBSE always asks to solve a system of 3 equations using matrix method. Set up AX = B, find |A|, find A⁻¹ by cofactors, then compute X = A⁻¹B.

Frequently asked questions

Are these Determinants notes free?

Yes — the Determinants notes for Mathematics (Class 12) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Determinants notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Determinants chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Determinants.