Class 12 Maths Chapter 4 Determinants Revision Notes for JEE Main & Advanced

1.0Cofactors of a Determinants of Matrix

Cofactor of an elements a_ijis expressed as C_ijand is calculated as:

C_ij= (–1)^i+j M_ij, where ‘i’ denotes the row and ‘j’ denotes the column.

Note: That a determinant of order 3 will have 9 minors, each minor will be a determinant of order 2 and a determinant of order4 will have 16 minors each minor will be a determinant of order 3.

Expansion of a determinant in Terms of the Elements of any Row or column:

$D = \sum_{j = 1}^{3} a_{ij} C_{ij} etc. for i = 1, 2, 3$

Sum of the product of elements of any row (or column) with cofactors of corresponding elements of any other row (column) is ZERO.

2.0Determinant Matrix Properties

Property 1: The value of a determinant stays the same if the row and columns are interchanged.

Property 2: If any 2 rows (or columns) of a matrix are interchanged, the determinant's value changes sign.

Property 3: If a determinant has any two identical rows (or columns), its value is zero.

If D= $a_{1} a_{1} a_{3} b_{1} b_{1} b_{3} c_{1} c_{1} c_{3} = 0$

Property 4: If all the elements of a row (or column) are multiplied by a scalar, the determinant is multiplied by that scalar.

If $D = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} D^{‘} = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}$

then $D^{'} = D D^{'} = D$

Note: $∣ KD ∣ = K^{n} .∣ D ∣$ where n is the order of determinants.

Property 5: If each element of a row (or column) is written as the sum of two terms, the determinant can be written as the sum of 2 separate determinants.

If $a_{1} + x a_{2} a_{3} b_{1} + y b_{2} b_{3} c_{1} + z c_{2} c_{3} = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} + x a_{2} a_{3} y b_{2} b_{3} z c_{2} c_{3}$

Property 6: The value of a determinant is not changed if a multiple of one row (or column) is added to another row (or column).

$Let D = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} and D^{'} = a_{1} + m a_{2} a_{2} a_{3} + n a_{1} b_{1} + m b_{2} b_{2} b_{3} + n b_{1} c_{1} + m c_{2} c_{2} c_{3} + n c_{1}$

These properties simplify the manipulation and calculation of determinants, making it easier to understand and solve complex matrix equations.

Also Read Determinant of 3x3 Matrix

3.0Determinants of Matrix Formula (SARRUS Method)

By SARRUS Method

D= $a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}$ =

= (a₁b₂c₃+ a₂b₃c₁ + a₃b₁c₂) – (a₃b₂c₁+ a₂b₁c₃+ a₁b₃c₂)

4.0Determinant of Square matrix

The determinant of a square matrix is a scalar quantity that reveals key information about the matrix and the linear transformation it embodies. For a matrix to be square, it must have the same number of rows and columns, such as 2 × 2, 3 × 3, 4 × 4, or in general, n × n.

Calculation Methods

For an n × n matrix A, the determinant, denoted as det(A) or $∣ A ∣$ can be calculated using various methods:

Cofactor Expansion:

This method involves expanding along any row or column using cofactors:

$det (A) = \sum_{j = 1}^{n} (- 1)^{i + j} \cdot a_{ij} \cdot det (M_{ij})$

Here, M_ij is the minor of element a_ij, calculated as the determinant of the (n – 1) × (n – 1) matrix obtained by removing the i^th row and i^th column from A.

Triangularization:

Transforming A into an upper or lower triangular matrix U or L, where the determinant is the product of the diagonal elements:

$det (A) = \prod_{i = 1}^{n} u_{ii} or det (A) = \prod_{i = 1}^{n} l_{ii}$

Eigenvalues:

If A has eigenvalues $λ_{1}, λ_{2}, \dots, λ_{n}$ , then:

$det (A) = λ_{1} \cdot λ_{2} \cdot \dots \cdot λ_{n}$

5.0Determinant of non-square matrix.

The determinant is a concept specifically defined for square matrices, meaning matrices with an equal number of rows and columns. For non-square matrices, the determinant is not defined. This is because many properties and operations that rely on the determinant, such as checking matrix invertibility or solving systems of linear equations using Cramer's rule, are inherently tied to the concept of a square matrix.

Key Points:

Non-Square Matrices: These matrices have a different number of rows and columns (e.g., m × n where m ≠ n).
Undefined Determinant: For non-square matrices, the determinant is not defined, as the fundamental properties and applications of the determinant do not apply to these matrices.

6.0Cramer’s Rule (System of Linear Equation)

(a) Equations involving two variables:

(i) Consistent Equations : Definite & unique solution (Intersecting lines)

(ii) Inconsistent Equations : No solution (Parallel lines)

(iii) Dependent Equations : Infinite solutions (Identical lines)

Let, a₁ x + b₁ y + c₁= 0

a₂ x + b₂ y + c₂ = 0 then:

(1) $\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}} \Rightarrow$ Given equations are consistent with unique solution

(2) $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}} \Rightarrow$ ⇒ Given equations are inconsistent

(3) $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}} \Rightarrow$ Given equations are consistent with infinite solutions

(b) Equations Involving Three variables:

Let a₁ x + b₁ y + c₁= d₁ ...(i)

a₂ x + b₂ y + c₂ = d₂ ...(ii)

a₃ x + b₃ y + c₃ = d₃ ...(iii)

Then, x = $\frac{D _{1}}{D}, y = \frac{D _{2}}{D}, z = \frac{D _{3}}{D}$ .

Where

$D = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}; D_{1} = d_{1} d_{2} d_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3};$

$D_{2} = a_{1} a_{2} a_{3} d_{1} d_{2} d_{3} c_{1} c_{2} c_{3} & D_{3} = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} d_{1} d_{2} d_{3}$

Note: (i) If D ≠ 0 and at least one of D₁, D₂, D₃≠ 0, then the given system of equations is consistent and has a unique non-trivial solution.

(ii) If D ≠ 0 & D₁ = D₂ = D₃ = 0, then the given system of equations is consistent and has trivial solutions only.

(iii) If D = 0but atleast one of D₁, D₂, D₃ is not zero then the equations are inconsistent and have no solution.

(iv) If D = D₁ = D₂ = D₃ = 0, then the given system of equations may have infinite or no solution.

Note: In case $a_{1} x + b_{1} y + c_{1} z = d_{1} a_{1} x + b_{1} y + c_{1} z = d_{2} a_{1} x + b_{1} y + c_{1} z = d_{3} ⎭ ⎬ ⎫$

(At Least two of d₁, d₂ & d₃ are not equal)

D = D₁ = D₂ = D₃ = 0. But these three equations represent three parallel planes. Hence the system is inconsistent.

7.0Determinant of Matrix Solved Example

Example 1: If $x + 3 3 - x 1 - 2 - 3 - 2 13 = 0$ , find .

Solution: $x + 3 3 - x 1 - 2 - 3 - 2 13$ =0

⇒ (x+3)[-6-(-3)]-1[9+x]-2[-9-2 x]

⇒ (x+3)(-6+3)-(9+x)-2(-9-2 x)=0

⇒ -3(x+3)-9-x+18+4 x=0

⇒ -3 x-9-9-x+18+4 x=0

⇒ -4 x+4 x-18+18=0

∴ x $\in$ R .

Example 2: If in the determinant $x 32 333 3 x 3, C_{11} = C_{22}$ , where C_{i j} is cofactor of element a_{i j} then x =

(A) 2 (B) –2 (C) $\frac{5}{2}$ (D) $- \frac{9}{8}$

Ans. (C)

Solution:

$C_{11} = C_{22} \Rightarrow (- 1)^{1 + 1} 33 x 3 = (- 1)^{2 + 2} x 2 33$

⇒ 9-3 x=3 x-6 $\Rightarrow$ -6 x=-15 $\Rightarrow$ x= $\frac{5}{2}$

Example 3: Find the value of the determinant $a^{2} ab a c ab b^{2} b c a c b c c^{2}$

Solution:

D= $a^{2} ab a c ab b^{2} b c a c b c c^{2} = a a ab a c b b^{2} b c c b c c^{2}$ = $ab c a a a b b b c c c$ =0

Example 4: $3^{2} + k 4^{2} + k 5^{2} + k 4^{2} 5^{2} 6^{2} 3^{2} + 3 + k 4^{2} + 4 + k 5^{2} + 5 + k = 0, then the value of k$

(A) 2 (B) 1 (C) –1 (D) 0

Ans. (B)

Solution:

Applying $(C_{3} \to C_{3} - C_{1}) D = 3^{2} + k 4^{2} + k 5^{2} + k 4^{2} 5^{2} 6^{2} 345$ =0

⇒ $9 + k 79 16911311 = 0 (R_{3} \to R_{3} - R_{2}; R_{2} \to R_{2} - R_{1})$

⇒ k-1=0 $\Rightarrow$ k=1

Example 5: Prove that $2336636655143112637$ =0

Solution:

$2336636655143112637$

Taking 11 common from C₂

= $112336636513112637 C_{1} \to C_{1} - (2 C_{2} + C_{3})$

= $110006513112637$ =0

Example 6: If α, β and γ are the roots of the equation x^3+p x+q=0 then the value of the determinant $α β γ β γ α γ α β$ is-

(A) p (B) q (C) p²– 2q (D) 0

Ans. (D)

Solution:

α, β , γ are roots of x³ + px + q = 0

∴ $α + β + γ$ =0

⇒ on $α β γ β γ α γ α β$ , Applying $C_{1} \to C_{1} + C_{2} + C_{3}$

$\Rightarrow (α + β + γ) 111 β γ α γ α β$

Example 7:

Using factor property of determinants prove that $111 x y z x^{2} y^{2} z^{2} = (x - y) (y - z) (z - x)$

Solution:

On putting x=y, y=z, z=x we are getting value of determinant is 0 so by factor theorems (x–y), (y–z) and (z–z) are factors. So, by factor theorem

$\Rightarrow 111 x y z x^{2} y^{2} z^{2} = λ (x - y) (y - z) (z - x)$

For λ put x = 0, y = 1, z = 2

We get λ = 1

So R.H.S. = (x-y)(y-z)(z-x)

Example 8:

Let $α & β$ be the roots of equation a x^2+b x+c=0 $and S_{n} = α^{n} + β^{n} for n \geq 1$ . Evaluate the value of the determinant $3 1 + S_{1} 1 + S_{2} 1 + S_{1} 1 + S_{2} 1 + S_{3} 1 + S_{2} 1 + S_{3} 1 + S_{4}$ .

Solution:

$3 1 + S_{1} 1 + S_{2} 1 + S_{1} 1 + S_{2} 1 + S_{3} 1 + S_{2} 1 + S_{3} 1 + S_{4} = 1 + 1 + 1 1 + a + b 1 + a^{2} + b^{2} 1 + a + b 1 + a^{2} + b^{2} 1 + a^{3} + b^{3} 1 + a^{2} + b^{2} 1 + a^{3} + b^{3} 1 + a^{4} + b^{4}$

= $111 1 a b 1 a^{2} b^{2} \times 111 1 a a^{2} 1 b b^{2} = 111 1 a b 1 a^{2} b^{2} = [(1 - a) (1 - b) (a - b)]^{2}$

$\Rightarrow D = (α - β)^{2} (α + β - α β - 1)^{2}$

∵ are roots of the equation

⇒ $a + b = \frac{- b}{a} ab = \frac{c}{a} \Rightarrow ∣ a - b ∣ = \frac{b ^{2} - 4 a c}{∣ a ∣}$

⇒ D= $\frac{\frac{( b ^{2} - 4 a c )}{a ^{2}} ( \frac{a + b + c}{a} ) ^{2}}{a} = \frac{( b ^{2} - 4 a c ) ( a + b + c ) ^{2}}{a ^{4}}$

Example 9: If the system of equations $x + λ y + 1 = 0, λ x + y + 1 = 0& x + y + λ = 0$ is consistent, then find the value of $λ$ .

Solution:

For consistency of the given system of equations

D= $1 λ 1 λ 11 11 λ$ =0

⇒ 3λ = 1 + 1 + λ³ or λ³ – 3λ + 2 = 0

⇒ (λ – 1)² (λ + 2) = 0

⇒ λ = 1 or λ = –2

8.0Determinant of Matrix Practice Question

1. $Let A = cos^{- 1} x cos^{- 1} y cos^{- 1} z cos^{- 1} y cos^{- 1} z cos^{- 1} x cos^{- 1} z cos^{- 1} x cos^{- 1} y$ such that |A|=0, then maximum value of x + y + z is

(A) 3 (B) 0 (C) 1 (D) 2

2. If α, β & γ are the roots of the equation x^3+p x+q=0, then the value of the determinant $α β γ β γ α γ α β$ =

(A)p (B) q (C) p² – 2q (D) none

3. If a, b & c are non-zero real numbers^, then D= $b^{2} c^{2} c^{2} a^{2} a^{2} b^{2} b c c a ab b + c c + a a + b$ =

(A)abc (B) a²b²c² (C) bc + ca + ab (D) zero

4. The determinant $b_{1} + c_{1} b_{2} + c_{2} b_{3} + c_{3} c_{1} + a_{1} c_{2} + a_{2} c_{3} + a_{3} a_{1} + b_{1} a_{2} + b_{2} a_{3} + b_{3}$ =

(A) $a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}$

(B) $2 a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}$

(D) none of these

5. The determinant $cos (θ + ϕ) sin θ - cos θ - sin (θ + ϕ) cos θ sin θ cos 2 ϕ sin ϕ cos ϕ$ is:

(A) 0 (B) independent of θ

Questions	1	2	3	4	5
Answers	A	D	D	B	B

9.0Sample Questions on Determinant of a Matrix

Q. How do you calculate the determinant of a 2 x 2 matrix?

Ans: For a 2×2 matrix A given by:

A= $(a c b d)$

The determinant is calculated as:

$det (A)$ =a d-b c

Q. How do you calculate the determinant of a 3 x 3 matrix?

Ans: For a 3 × 3 matrix A given by:

A= $a d g b e h c f i$

The determinant is calculated using the cofactor expansion along the first row:

$det (A) = a (e i - f h) - b (d i - f g) + c (d h - e g)$

Q. How do you find the determinant of a triangular matrix?

Ans: In both upper and lower triangular matrices, the determinant equals the product of their diagonal elements. For an n × n triangular matrix T with diagonal elements $t_{11}, t_{22}, \dots, t_{nn}$ :

$det (T) = t_{11} \cdot t_{22} \cdot \dots \cdot t_{n}$

Q. What are some properties of the determinant?

Ans:

Multiplicative Property: $det (A B) = det (A) \cdot det (B)$
Transpose Property: $det (A^{T}) = det (A)$
Inverse Property: If A is invertible, $det (A^{- 1}) = \frac{1}{det ( A )}$

Row Operations: Swapping two rows of a matrix multiplies the determinant by –1, scaling a row by a factor k multiplies the determinant by k, and adding a multiple of one row to another does not change the determinant.

Q. How does the determinant relate to eigenvalues?

Ans: For a square matrix A, the determinant is the product of its eigenvalues $λ_{1}, λ_{2}, \dots, λ_{n} : det (A) = λ_{1} \cdot λ_{2} \cdot \dots \cdot λ_{n}$

Q. How does one use the determinant to solve a system of linear equations?

Ans: Cramer's Rule uses the determinant to solve a system of linear equations AX = B. If A is an n × n matrix with $det (A) \neq = 0$ , the solution for the variables x_i can be found using:

$x_{i} = \frac{det ( A _{i} )}{det ( A )}$ , where A_i is the matrix obtained by substituting the i^th column of A with the column vector B.

Parameter	Details
Difficulty Level	Moderate to High
Key Concepts Covered	Types of relations and functions, algebra of functions, binary operations
Revision Time	20–25 Minutes

Topic	Overview
Determinants	Definition and notation
Properties of Determinants	Standard simplification properties
Minors and Cofactors	Concepts and computation
Adjoint of a Matrix	Definition and construction
Inverse Using Adjoint	Finding inverse through determinants
Elementary Operations	Effect on determinant value
System of Linear Equations	Solution using Cramer's Rule
Applications	Area, algebra, and linear systems

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Test Paper with Answers

Topic	JEE Importance	Difficulty Level
Properties of Determinants	Very High	Moderate
Minors & Cofactors	High	Moderate
Adjoint & Inverse	Very High	Moderate
Elementary Transformations	High	Moderate
Cramer's Rule	High	Moderate
Applications of Determinants	High	Moderate

Common Mistake	Correct Approach
Incorrect sign while expanding determinants	Follow the cofactor sign pattern carefully: (+,-,+;,-,+,-;,+,-,+).
Confusing minors and cofactors	Remember that the cofactor includes the sign factor ((-1)^{i+j}).
Applying determinant properties incorrectly	Revise each property before simplifying lengthy determinants.
Forgetting that swapping two rows changes the sign	Reverse the sign of the determinant whenever two rows or columns are interchanged.
Calculating the inverse without checking the determinant	Ensure that (
Errors while finding the adjoint	First compute all cofactors correctly, then take their transpose.
Ignoring elementary transformation shortcuts	Use row and column operations to simplify lengthy calculations.
Arithmetic mistakes in expansion	Simplify step by step and verify intermediate calculations.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Properties of Determinants	2–3	Moderate
Minors & Cofactors	1–2	Moderate
Adjoint & Inverse	2–3	Moderate
Elementary Transformations	1–2	Moderate
Cramer's Rule	1–2	Moderate
Mixed Determinant Problems	2–3	Moderate–Difficult

Test your Knowledge

question 1 of 5

A determinant is defined only for a:

Frequently Asked Questions

What are the most common mistakes students make while solving Determinant questions?

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Class 12 Maths Chapter 4 Determinants

1.0Chapter Snapshot

2.0Chapter 4 Smart Study Material by ALLEN Experts: Download PDFs

Test your Knowledge

question 1 of 5

A determinant is defined only for a:

Determinant of a Matrix

1.0Definition of Determinant of Matrix

2.0Matrix Minors of a Determinants

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

Frequently Asked Questions

What are the most common mistakes students make while solving Determinant questions?

Why are Determinants important for JEE Main and JEE Advanced?

How can I revise the Determinants chapter quickly before the exam?

Which chapter should I study after completing Determinants?

Which topics from Determinants are most frequently asked in JEE?

Join ALLEN!

Class 12 Maths Chapter 4 Determinants

1.0Chapter Snapshot

2.0Chapter 4 Smart Study Material by ALLEN Experts: Download PDFs

Determinant of a Matrix

1.0Definition of Determinant of Matrix

2.0Matrix Minors of a Determinants

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy