Class 12 Mathematics Notes Chapter 4 (Determinants) – Examplar Problems (English) Book

Alright class, let's get straight into Chapter 4: Determinants. This is a crucial chapter, not just for your board exams but also for various government entrance exams where Mathematics is a component. We'll focus on the core concepts, properties, and applications, keeping the competitive exam perspective in mind.

Chapter 4: Determinants - Detailed Notes for Government Exam Preparation

1. Introduction

• Definition: A determinant is a unique number (scalar value) associated with every square matrix. It encapsulates certain properties of the matrix and the linear transformation it represents.
• Notation: The determinant of a square matrix A is denoted by det(A) or |A| or Δ.
• Important: Determinants exist ONLY for square matrices.

2. Determinant of a Matrix

• (a) Order 1: If A = [a], then |A| = a.
• (b) Order 2: If A = [[a, b], [c, d]], then |A| = ad - bc.
• (c) Order 3: If A = [[a11, a12, a13], [a21, a22, a23], [a31, a32, a33]]
  • Expansion along Row 1 (R1):
    |A| = a₁₁ * C₁₁ + a₁₂ * C₁₂ + a₁₃ * C₁₃
    Where Cᵢⱼ is the Cofactor of the element aᵢⱼ.
  • Expansion along any Row (Rᵢ) or Column (Cⱼ): The determinant value is the sum of the product of elements of any one row (or column) with their corresponding cofactors.
    |A| = Σ (aᵢⱼ * Cᵢⱼ) for a fixed row 'i' or a fixed column 'j'.
  • Sign Convention for Cofactors: The sign (-1)ⁱ⁺ʲ is crucial. Remember the checkerboard pattern:
    [[+, -, +], [-, +, -], [+, -, +]]
  • Sarrus' Rule (Only for 3x3): A shortcut method. Rewrite the first two columns to the right of the determinant. Sum the products of the main diagonals (top-left to bottom-right) and subtract the sum of the products of the anti-diagonals (top-right to bottom-left). While quick, understanding expansion via cofactors is more fundamental.

3. Minors and Cofactors

• Minor (Mᵢⱼ): The minor of an element aᵢⱼ in a determinant is the determinant obtained by deleting the i-th row and j-th column.
• Cofactor (Cᵢⱼ or Aᵢⱼ): The cofactor of an element aᵢⱼ is defined as Cᵢⱼ = (-1)ⁱ⁺ʲ * Mᵢⱼ.
• Important Property: The sum of the product of elements of one row (or column) with the cofactors of another row (or column) is always zero.
  Example: a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C₂₃ = 0.

4. Properties of Determinants (Extremely Important for Competitive Exams)

These properties help simplify calculations significantly.

• (P1) Reflection Property: The value of the determinant remains unchanged if its rows and columns are interchanged. |A| = |Aᵀ|.
• (P2) All-Zero Property: If all elements of a row (or column) are zero, then the value of the determinant is zero.
• (P3) Proportionality/Identical Property: If any two rows (or columns) are identical or proportional (one is a scalar multiple of the other), then the value of the determinant is zero.
• (P4) Switching Property: If any two rows (or columns) are interchanged, the sign of the determinant changes.
• (P5) Scalar Multiple Property: If each element of a row (or column) is multiplied by a constant k, then the value of the new determinant is k times the original determinant.
  • Corollary: If A is an n x n matrix, then |kA| = kⁿ |A|. (Very Important!)
• (P6) Sum Property: If some or all elements of a row (or column) are expressed as the sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants.
  Example: |[[a+x, b+y], [c, d]]| = |[[a, b], [c, d]]| + |[[x, y], [c, d]]|
• (P7) Invariance Property (Row/Column Operations): Applying operations like Rᵢ → Rᵢ + kRⱼ or Cᵢ → Cᵢ + kCⱼ does not change the value of the determinant. (This is the most used property for simplification).

5. Area of a Triangle

• The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by:
  Area (Δ) = ½ | x₁ (y₂ - y₃) + x₂ (y₃ - y₁) + x₃ (y₁ - y₂) |
• Determinant Form:
  Area (Δ) = ½ | [[x1, y1, 1], [x2, y2, 1], [x3, y3, 1]] | (Take the absolute value of the result).
• Condition for Collinearity: Three points are collinear if the area of the triangle formed by them is zero.
  [[x1, y1, 1], [x2, y2, 1], [x3, y3, 1]] = 0

6. Adjoint of a Square Matrix

• Definition: The adjoint of a square matrix A, denoted by adj(A), is the transpose of the matrix of cofactors of the elements of A.
  If A = [aᵢⱼ], let C = [Cᵢⱼ] be the matrix of cofactors. Then adj(A) = Cᵀ.
• Key Theorem: For any square matrix A of order n:
  A (adj A) = (adj A) A = |A| Iₙ
  where Iₙ is the identity matrix of order n.
• Properties of Adjoint:
  • If A is a square matrix of order n, then |adj A| = |A|ⁿ⁻¹
  • adj(AB) = (adj B)(adj A)
  • adj(Aᵀ) = (adj A)ᵀ
  • adj(kA) = kⁿ⁻¹ adj(A) (where k is a scalar)
  • adj(adj A) = |A|ⁿ⁻² A (if |A| ≠ 0)

7. Inverse of a Square Matrix

• Singular Matrix: A square matrix A is singular if |A| = 0.
• Non-Singular Matrix: A square matrix A is non-singular if |A| ≠ 0.
• Invertible Matrix: A square matrix A is invertible (i.e., its inverse exists) if and only if it is non-singular (|A| ≠ 0).
• Formula for Inverse:
  A⁻¹ = (1 / |A|) * adj(A)
• Properties of Inverse:
  • (A⁻¹)⁻¹ = A
  • (AB)⁻¹ = B⁻¹ A⁻¹ (Reversal Law)
  • (Aᵀ)⁻¹ = (A⁻¹)ᵀ
  • |A⁻¹| = 1 / |A| = |A|⁻¹

8. Applications: Solving Systems of Linear Equations

Consider a system of n linear equations in n variables:
a₁₁x₁ + a₁₂x₂ + ... + a₁nxn = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂nxn = b₂
...
a<0xE2><0x82><0x99>₁x₁ + a<0xE2><0x82><0x99>₂x₂ + ... + a<0xE2><0x82><0x99>nxn = b<0xE2><0x82><0x99>

This system can be written in matrix form as AX = B, where:
A = [[a11, ..., a1n], ..., [an1, ..., ann]] (Coefficient Matrix)
X = [[x1], ..., [xn]] (Variable Matrix)
B = [[b1], ..., [bn]] (Constant Matrix)

• Solution using Matrix Inverse (Matrix Method):

  • Calculate |A|.
  • Case 1: |A| ≠ 0 (Non-singular Matrix)
    The system has a unique solution given by X = A⁻¹B.
    To find X, calculate A⁻¹ = (1/|A|) adj(A) and multiply by B.
  • Case 2: |A| = 0 (Singular Matrix)
    Calculate (adj A)B.
    • Subcase 2a: (adj A)B ≠ O (Zero Matrix)
      The system has no solution (Inconsistent).
    • Subcase 2b: (adj A)B = O (Zero Matrix)
      The system may have infinitely many solutions or no solution. It is generally considered consistent with infinitely many solutions in most contexts, but further investigation might be needed depending on the specific system. (For most competitive exams, this case implies infinitely many solutions if the system is stated to be consistent).

• Consistency:

  • A system is consistent if it has one or more solutions (unique or infinite).
  • A system is inconsistent if it has no solution.

• Homogeneous System (AX = O): If B = O, the system is homogeneous.

  • If |A| ≠ 0, it has only the trivial solution (X = O, i.e., x₁=x₂=...=x<0xE2><0x82><0x99>=0).
  • If |A| = 0, it has infinitely many non-trivial solutions.

Multiple Choice Questions (MCQs)

1. If A is a 3x3 matrix such that |A| = 5, then what is the value of |2A|?
   (a) 10
   (b) 20
   (c) 40
   (d) 80

2. The value of the determinant |[[1, a, b+c], [1, b, c+a], [1, c, a+b]]| is:
   (a) a+b+c
   (b) 1
   (c) 0
   (d) abc

3. If the points (2, -3), (k, -1), and (0, 4) are collinear, then the value of k is:
   (a) 4/5
   (b) 5/4
   (c) 10/7
   (d) 7/10

4. If A is a square matrix of order 3 and |A| = -4, then |adj A| is equal to:
   (a) -4
   (b) 4
   (c) -16
   (d) 16

5. If A = [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]], then A⁻¹ is equal to:
   (a) [[cos(theta), sin(theta)], [-sin(theta), cos(theta)]]
   (b) [[-cos(theta), sin(theta)], [sin(theta), cos(theta)]]
   (c) [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]
   (d) [[-cos(theta), -sin(theta)], [-sin(theta), -cos(theta)]]

6. A square matrix A is said to be singular if:
   (a) |A| = 1
   (b) |A| ≠ 0
   (c) |A| = 0
   (d) adj A = O

7. For the system of equations x + 2y = 5 and 3x + ky = 15 to have infinitely many solutions, the value of k must be:
   (a) 2
   (b) 3
   (c) 6
   (d) 5

8. If A = [[2, 0], [0, 3]], then A(adj A) is equal to:
   (a) [[6, 0], [0, 6]]
   (b) [[3, 0], [0, 2]]
   (c) [[1/2, 0], [0, 1/3]]
   (d) [[0, 6], [6, 0]]

9. If |[[x, 2], [18, x]]| = |[[6, 2], [18, 6]]|, then x is equal to:
   (a) 6
   (b) ±6
   (c) -6
   (d) 0

10. If A is an invertible matrix of order 2, then det(A⁻¹) is equal to:
    (a) det(A)
    (b) 1 / det(A)
    (c) 1
    (d) 0

Answer Key for MCQs:

1. (c) 40 [Using |kA| = kⁿ|A|, |2A| = 2³|A| = 8 * 5 = 40]
2. (c) 0 [Apply C₃ → C₃ + C₂. Then C₃ becomes (a+b+c) which can be taken common. Then C₁ and C₃ become proportional (C₃ = (a+b+c)C₁), hence determinant is 0.]
3. (c) 10/7 [Area = 0. ½ |[2(-1-4) + k(4-(-3)) + 0(-3-(-1))]| = 0 => |-10 + 7k| = 0 => 7k = 10 => k = 10/7]
4. (d) 16 [Using |adj A| = |A|ⁿ⁻¹, |adj A| = (-4)³⁻¹ = (-4)² = 16]
5. (a) [[cos(theta), sin(theta)], [-sin(theta), cos(theta)]] [Calculate |A| = cos²θ - (-sin²θ) = 1. adj A = [[cos(theta), sin(theta)], [-sin(theta), cos(theta)]]. A⁻¹ = (1/|A|)adj A = adj A.]
6. (c) |A| = 0 [Definition of a singular matrix]
7. (c) 6 [For infinite solutions in a₂x + b₂y = c₂, a₁x + b₁y = c₁, we need a₁/a₂ = b₁/b₂ = c₁/c₂. Here 1/3 = 2/k = 5/15. From 1/3 = 2/k, we get k=6. Check: 5/15 = 1/3. Condition holds.]
8. (a) [[6, 0], [0, 6]] [Using A(adj A) = |A|I. Here |A| = (2)(3) - (0)(0) = 6. So A(adj A) = 6 * [[1, 0], [0, 1]] = [[6, 0], [0, 6]]]
9. (b) ±6 [x² - 36 = 36 - 36 => x² - 36 = 0 => x² = 36 => x = ±6]
10. (b) 1 / det(A) [Property: |A⁻¹| = 1 / |A|]

Study these concepts thoroughly, especially the properties and the conditions for solving systems of equations. Practice problems from your NCERT Exemplar book to solidify your understanding. Good luck!