Determinants
Complete Class XII Mathematics chapter notes covering determinants, minors, cofactors, properties, area of a triangle, adjoint and inverse of a matrix, Cramer's Rule, and the complete conditions for solutions of linear equations.
Foundation
Introduction to Determinants
A determinant is a scalar value associated with a square matrix. Determinants provide important information about a matrix and are used to find inverses, solve systems of linear equations, determine areas and study the existence and uniqueness of solutions.
| a | b |
| c | d |
Fundamentals
Determinant Basics
Square matrices only
A determinant is defined only for a square matrix. A matrix having different numbers of rows and columns does not have a determinant in the usual Class XII sense.
Order
An n × n determinant contains n rows and n columns. Therefore, its order is n.
3 × 3 → third order
Important distinction
A matrix is an arrangement of elements enclosed in square brackets, while a determinant is a scalar expression represented using vertical bars.
Evaluation
Determinants of Order 2 and 3
Second-order determinant
| a | b |
| c | d |
Multiply the entries on the principal diagonal, multiply the entries on the other diagonal, and subtract the second product from the first.
Third-order determinant
| a | b | c |
| d | e | f |
| g | h | i |
A third-order determinant can be expanded along any row or any column using minors and cofactors.
Expansion along the first row
Core Concept
Minors and Cofactors
Minor
The minor Mij of an element aij is the determinant obtained after deleting the i-th row and j-th column containing that element.
Cofactor
The cofactor Cij is obtained by multiplying the minor by the sign factor (−1)i+j.
Sign pattern of cofactors
The signs alternate like a checkerboard.
Expansion
Expansion of a Determinant
A determinant can be expanded along any row or any column. The value of the determinant remains unchanged when the expansion is performed correctly.
Important Results
Properties of Determinants
Interchange of two rows
If any two rows of a determinant are interchanged, the value of the determinant changes its sign.
Interchange of two columns
If any two columns are interchanged, the determinant changes its sign.
Equal rows or columns
If two rows or two columns are identical, the determinant is zero.
Proportional rows or columns
If two rows or two columns are proportional, the determinant is zero.
Factor from a row or column
A common factor in the elements of a row or column can be taken outside the determinant.
Adding a multiple
Adding a multiple of one row to another row does not change the value of the determinant.
Transpose
The determinant of a matrix is equal to the determinant of its transpose.
Triangular determinant
For a triangular determinant, its value is the product of the diagonal elements.
High-value property
The determinant remains unchanged on taking transpose.
Application
Area of a Triangle Using Determinants
If the coordinates of three points are (x₁, y₁), (x₂, y₂) and (x₃, y₃), then the area of the triangle is:
x₂ y₂ 1
x₃ y₃ 1 |
Collinearity condition
Three points are collinear if and only if the determinant representing their area is zero.
Inverse
Adjoint and Inverse of a Matrix
Adjoint
The adjoint of a square matrix A is the transpose of its cofactor matrix.
Inverse formula
The inverse exists only when |A| ≠ 0.
Key condition
A non-singular square matrix has a unique inverse.
Inverse of a 2 × 2 matrix
| a | b |
| c | d |
A−1 = 1/ (ad − bc) [
| d | −b |
| −c | a |
provided ad − bc ≠ 0.
Application
Linear Equations in Matrix Form
A system of linear equations can be written compactly in matrix form as:
A
Coefficient matrix
X
Column matrix of variables
B
Constant matrix
When A is invertible
A−1AX = A−1B
X = A−1B
Very Important
Conditions of Solutions of Linear Equations
General system of three linear equations
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Coefficient determinant
| a₁ | b₁ | c₁ |
| a₂ | b₂ | c₂ |
| a₃ | b₃ | c₃ |
Augmented determinant concept
To study consistency in a general system, compare the rank of the coefficient matrix with the rank of the augmented matrix.
Complete Conditions
Conditions for consistency and number of solutions.
| Condition | Nature of System | Solution |
|---|---|---|
| ρ(A) = ρ([A|B]) = n | Consistent | Unique solution |
| ρ(A) = ρ([A|B]) < n | Consistent | Infinitely many solutions |
| ρ(A) ≠ ρ([A|B]) | Inconsistent | No solution |
Determinant-based conditions for three equations
Unique Solution
If the coefficient determinant is non-zero, the coefficient matrix is non-singular and the system has a unique solution.
Δ = 0
The determinant test alone does not decide whether the system has no solution or infinitely many solutions. Further consistency conditions are required.
Δ = 0: Further Test
Compare the relevant determinants or use rank conditions to distinguish between infinitely many solutions and no solution.
Cramer's Rule — solution conditions
Unique solution exists.
Cramer's formulas with division by Δ cannot be directly used; consistency must be checked.
The system has infinitely many solutions, provided the coefficient system is rank-deficient but consistent.
The system is inconsistent and has no solution.
Two linear equations in two variables
a₂x + b₂y + c₂ = 0
| Condition | Interpretation | Solution |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Lines intersect at one point | Unique solution |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel distinct lines | No solution |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident lines | Infinitely many solutions |
Method
Cramer's Rule
For a system of three linear equations, define Δ as the determinant of the coefficient matrix. Δ₁, Δ₂ and Δ₃ are obtained by replacing the first, second and third columns respectively by the constants.
These formulas apply when Δ ≠ 0.
Worked Problems
Solved Examples
Example 1 — Evaluate a second-order determinant
| 4 | 7 |
| 2 | 5 |
Example 2 — Evaluate a third-order determinant
| 1 | 2 | 3 |
| 0 | 1 | 4 |
| 5 | 6 | 0 |
= −24 + 40 − 15
= 1
Example 3 — Find a minor
| 2 | 3 | 1 |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
To find M₂₃, delete row 2 and column 3.
| 2 | 3 |
| 7 | 8 |
Example 4 — Find a cofactor
Example 5 — Find area of a triangle
3 4 1
5 1 1 |
= ½ |3 + 4 − 17|
= ½ × 10
= 5 square units
Example 6 — Find inverse using 2 × 2 formula
| 2 | 3 |
| 1 | 2 |
| 2 | −3 |
| −1 | 2 |
Example 7 — Solve two equations using determinants
x − y = 1
| 2 | 1 |
| 1 | −1 |
| 5 | 1 |
| 1 | −1 |
| 2 | 5 |
| 1 | 1 |
y = Δ₂/Δ = 1
Example 8 — Determine the nature of a system
2x + 2y = 4
The equations represent the same line.
Example 9 — Identify no solution
2x + 2y = 5
Example 10 — Determine unique solution using Δ
Example 11 — Verify a determinant property
| 1 | 2 |
| 3 | 4 |
| 3 | 4 |
| 1 | 2 |
Example 12 — Prove determinant is zero
| 1 | 2 | 3 |
| 2 | 4 | 6 |
| 4 | 5 | 7 |
Example 13 — Determine whether inverse exists
| 3 | 2 |
| 6 | 4 |
Example 14 — Three-equation system with unique solution
x + 2y + 3z = 14
2x + y + z = 7
| 1 | 1 | 1 |
| 1 | 2 | 3 |
| 2 | 1 | 1 |
Example 15 — Infinite solutions using rank condition
Example 16 — No solution using rank condition
ρ([A|B]) = 3
Self Assessment
Practice Questions
Q1. Evaluate the determinant | 5 3 ; 2 4 |.
Q2. Evaluate a third-order determinant using expansion along the first row.
Q3. Find M₂₃ for the matrix [ [1,2,3], [4,5,6], [7,8,9] ].
Q4. Find C₁₂ if M₁₂ = 7.
Q5. State the effect of interchanging two rows of a determinant.
Q6. What happens if two rows of a determinant are identical?
Q7. Find the area of a triangle using the determinant formula.
Q8. When does the inverse of a square matrix exist?
Q9. Write the inverse formula using adjoint.
Q10. What is the condition for a unique solution of AX = B?
Q11. What is the rank condition for infinitely many solutions?
Q12. What is the rank condition for no solution?
Q13. If Δ = 0, can Cramer's Rule directly give x, y and z?
Q14. For two equations, state the condition for a unique solution.
Q15. For two equations, state the condition for no solution.
Q16. For two equations, state the condition for infinitely many solutions.
Q17. Show that |A| = |Aᵀ|.
Q18. What is the determinant of a triangular matrix?
Q19. If one row is a multiple of another row, what is the determinant?
Q20. If ρ(A)=ρ([A|B])=3 for a system with three variables, what type of solution exists?
Higher Order Thinking
HOT Questions
If A is a square matrix and |A| = 0, explain why A⁻¹ does not exist.
Prove that if two rows of a determinant are proportional, then its value is zero.
Show that the determinant of a matrix is equal to the determinant of its transpose.
If Δ = 0 for a system of three equations, explain why another test is necessary.
For a system of three equations, explain the difference between Δ = 0, Δ₁ = Δ₂ = Δ₃ = 0 and the case where at least one of Δ₁, Δ₂, Δ₃ is non-zero.
Derive the inverse formula for a 2 × 2 matrix.
Explain geometrically why coincident lines give infinitely many solutions.
Explain geometrically why parallel distinct lines give no solution.
If ρ(A)=ρ([A|B])=2<3, explain why one obtains infinitely many solutions.
Solve a three-variable system using Cramer's Rule and verify the solution.
Last Minute Revision
Quick Revision Sheet
Exam Strategy
Exam Tips & Common Mistakes
Do this
Avoid these mistakes
Chapter 4 Final Checklist
Δ ≠ 0 → Unique Solution
ρ(A) = ρ([A|B]) < n → Infinitely Many Solutions
ρ(A) ≠ ρ([A|B]) → No Solution
Strengthen Your Determinants Concepts
Build confidence with structured chapter notes, practice questions, solved examples and focused preparation for linear equations and determinant-based problems.