JEE Main Study Notes on Properties of Determinants: Key Properties, Important Formulas, Study Tips & Previous Year Solved Questions

JEE Main Study Notes on Properties of Determinants: A determinant can be determined from a square matrix. Say, D is denoted det (D), |D| or det D. Properties of Determinants help to easily find the same value of the determinants with different and simpler configurations of elements.

• The topic Properties of Determinants have a weightage of 6-7% in JEE Main.
• JEE Main Study Notes on Properties of Determinants cover important properties like Reflection Property, All Zero Property, Proportionality, Scalar Multiple Property, Sum Property, etc.

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Basic Properties of A Determinant

Some basic properties of determinants are given below: det (A+B+C) + det C ≥ det (A+B) + det (B+C)

1. If I is the identity matrix of the order m ×m, then det(I) is equal to1 ; | I |=1
2. If the matrix XT is the transpose of matrix X, then det (XT) = det (X)
3. If matrix X-1 is the inverse of matrix X, then det (X-1) = 1/det (x) = det(X)-1
4. If two square matrices x and y are of equal size, then det (XY) = det (X) det (Y)
5. If matrix X retains size a × a and C is a constant, then det (CX) = Ca det (X)
6. If A, B, and C are three positive semidefinite matrices of equal size, then the following equation holds along with the corollary det (A+B) ≥ det(A) + det (B) for A, B, C ≥ 0;
7. In a triangular matrix, the determinant is equal to the product of the diagonal elements.
8. The determinant of a matrix is zero if each element of the matrix is equal to zero.
9. Laplace’s Formula and the Adjugate Matrix.

Check: JEE Main Study Notes for Probability

Important properties of Determinants

• Reflection Property: A determinant remains unaltered in its numerical value if the rows and columns are interchanged.
• Switching Property: If two parallel rows (or columns) are interchanged, then the determinant retains its numerical value but changes its sign.
• All- Zero Property: The determinants will be equivalent to zero if each term of rows and columns is zero.
• Proportionality (Repetition Property): A determinant is zero if any two parallel rows (or columns) are proportional.
• Scalar Multiple Property: If each element of a row (or column is multiplied by the same factor, the whole determinant is multiplied by the same factor.
• Sum Property: If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of the determinants.

        

• Triangle Property: If each term of a determinant above or below the main diagonal comprises zeroes, then the determinant is equivalent to the product of diagonal terms. That is

        

• Property of Invariance: The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column)
  It implies that the determinant remains unchanged under an operation of the term Ci⟶ Ci + αCj + βCkj where j and k are not equivalent to i, or a Mathematical operation of the term Ri ⟶ Ri + αRj + βRk, where, j and k are not equivalent to i.

         

• Factor property: If a determinant D becomes zero on putting x=α, then we say that (x−α) is a factor of determinant.
• Determinant of Cofactor Matrix:

       
           In the above determinants of the cofactor matrix, cij denotes the cofactor of the elements aij in Δ.

• Minors and Cofactors of Determinant:

A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration. Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij. The cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by A = (–1)i+j M, where M is minor of aij.

Must Read:

• JEE Main Study Notes for Trigonometry
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JEE Main Determinants: Important Tips for Preparation

• Positive Mental Set- The first step should be to cultivate a good attitude toward the subject. Stop avoiding mathematics as a subject. Be self-assured and open to new experiences.
• Start early- Students should begin their JEE Main preparations as soon as possible. Students usually begin in the eleventh grade. During this time, they will go over JEE Math syllabus for JEE Main exams. Learn about the crucial topics and plan your preparation accordingly.
• Practice- Students can begin by learning the fundamental ideas and then go to more complex concepts such as hyperbola and the binomial theorem. Students won't be able to learn Math ideas by simply reading about them or following the steps to solve problems. They should not begin solving problems until they have fully understood the concept. At the same time, they must effectively manage their time while doing so.

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• Avoid Rote Learning- When it comes to math, memorizing concepts or rote learning should be avoided. This subject necessitates stronger practical and analytical abilities. It's critical that you grasp the concepts completely and have answers to both the "How" and "Why" questions.
• Math Dictionary- While Math involves a lot of formulae, keep a separate notebook for them. You can write all of them down in this notebook and review them later. Students can also construct or keep a math dictionary and flashcards to help them revise or remember formulas fast and effortlessly at any time.

Also Check: JEE Main 2022 Best Preparation Books

Time Required for Preparation and Difficulty Level

Properties of Determinants is a sub-topic of Matrices and Determinants. It can take up to 2-3 weeks to complete the chapter. The chapter is of high importance and candidates must not avoid it. With proper preparation and planning, Candidates can score well. Check JEE Main 2022 Mathematics preparation

Also Check : JEE Main 2022 Preparation Apps

Previous Year Questions With Solutions

Question 1. If P=is the adjoint of a 3×3 matrix A and |A|=4, then α is equal to :

• 4
• 11
• 5
• 0 [JEE-2013]

Solution: B. |P|=1(12−12)−α(4−6)+ 3(4−6)=2α−6

Now, adjA=P

⇒|adjA|=|P|

⇒|A|²=|P|

⇒|P|=16

⇒2α−6=16

⇒α=11

Question 2. If ????, ???? ⍯ 0, and f(n)=????n+????n and If K(1-????)2(1-???? )2(????-????)2, then K is equal to:

1. 1
2. -1
3. ????????
4. 1/???????? [JEE-2014 ]

Solution: A. Consider,

So, k=1.

Question 3.The set of all values of λ for which the system of linear equations: 2x₁-2x₂+x₃=ƛx₁ 2x₁-3x₂+2x₃=ƛx₂ -x₁+2x₂=ƛx₃ has a non-trivial solution

• Contains two elements
• Contains more than two elements
• In an empty set
• Singleton [JEE-2015 ]

Solution: A. 2x₁-2x₂+x₃=ƛx₁

2x₁-3x₂+2x₃=ƛx₂

-x₁+2x₂=ƛx₃

(2-ƛ)x₁-2x₂+x₃=0

2x₁-(3-ƛ)x₂+2x₃=0

-x₁+2x₂-ƛx₃=0

For a non-trivial solution,

Δ=0 Hence ƛ has 2 values.

Question 4.The system of linear equations x+λy−z=0 λx−y−z=0 x+y−λz=0 has a non-trivial solution for:

• Exactly two values of λ
• Exactly three values of λ
• Infinitely many values of λ
• Exactly one value of λ [JEE-2016 ]

Solution: B. For trivial solutions,

⇒ −λ (λ+1)(λ+1)=0

⇒ λ=0, +1,−1

Question 5.The number of distinct real roots of the equation,

• 4
• 3
• 2
• 1 [JEE-2014 ]

Solution: C. Given,

∴ Number of solutions = 2

Question 6. If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :

• An empty set
• An infinite set
• A finite set containing two or more elements
• A singleton [JEE-2017]

Solution: D.

⇒ 1 [a – b] – 1 [1 – a] + 1 [b – a2] = 0

⇒ (a - 1)2 = 0

⇒ a = 1 For a = 1,

the equations become x + y + z = 1 x + y + z = 1 x + by + z = 0

These equations give no solution for b = 1

⇒ S is a singleton set.

Question 7.then the ordered pair (A, B) is equal to

• (4,5)
• (-4,-5)
• (-4,3)
• (-4,5) [JEE-2018]

Solution: D.

By comparing both sides we get, A = −4 and B = 5

Check:

• JEE Main Study notes for Integral Calculus
• JEE Main Study Notes on Differential Calculus

JEE Main Study Notes on Properties of Determinants FAQs

Question. How many questions will be asked from properties of determinants in JEE Main 2022?

Answer. Properties of determinants are a very important topic and are given high weightage. The expected number of questions asked on this topic in JEE Main 2022 is 2-3.

Question. What are the topics included under Matrices and Determinants in JEE Main 2022 Mathematics Syllabus?

Answer. The Matrices and Determinants include Matrices: Algebra of matrices, types of matrices, and matrices of order two and three; Determinants: Properties of determinants, evaluation of determinants, the area of triangles using determinants; Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations; Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices. For more details visit: JEE Main Mathematics Syllabus

Question. What is the Scalar Property of Determinants?

Answer. Scalar property of a determinant dictates that : “A determinant remains unaltered in its numerical value if the rows and columns are interchanged.”

Question: What is a cofactor of a Determinant?

Answer. A cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by A = (–1)i+j M, where M is minor of aij. A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration.

Question. How much time will be needed to prepare Determinants for JEE Main 2022?

Answer. Properties of Determinants is a subtopic of Matrices and Determinants. It can take up to 2-3 weeks to complete the chapter for JEE Main 2022. The chapter is of high importance and candidates must not avoid it.

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