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Class 12 Science (PCM)

Mathematics Part 1

Determinants

Determinants

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Row and Column Operations on Determinants
Accuracy: 37%
Total Attempted38
Total Correct14
Total Incorrect24
Difficulty Level:
High
Cofactor Expansion Theorem
Accuracy: 49%
Total Attempted39
Total Correct19
Total Incorrect20
Difficulty Level:
High
Applications of Determinants in Solving Linear Equations
Accuracy: 51%
Total Attempted55
Total Correct28
Total Incorrect27
Difficulty Level:
Medium
Determinant of a Matrix
Accuracy: 63%
Total Attempted57
Total Correct36
Total Incorrect21
Difficulty Level:
Medium
Zero Determinant Conditions
Accuracy: 66%
Total Attempted32
Total Correct21
Total Incorrect11
Difficulty Level:
Medium
Determinant of Order Three
Accuracy: 72%
Total Attempted25
Total Correct18
Total Incorrect7
Difficulty Level:
Medium
Expansion of Determinants
Accuracy: 75%
Total Attempted28
Total Correct21
Total Incorrect7
Difficulty Level:
Low
Collinearity Using Determinants
Accuracy: 76%
Total Attempted17
Total Correct13
Total Incorrect4
Difficulty Level:
Low
Effects of Interchanging Rows or Columns
Accuracy: 77%
Total Attempted30
Total Correct23
Total Incorrect7
Difficulty Level:
Low
Equation of a Line Using Determinants
Accuracy: 78%
Total Attempted9
Total Correct7
Total Incorrect2
Difficulty Level:
Low
Adjoint and Inverse of a Matrix
Accuracy: 81%
Total Attempted43
Total Correct35
Total Incorrect8
Difficulty Level:
Low
Minors and Cofactors
Accuracy: 82%
Total Attempted28
Total Correct23
Total Incorrect5
Difficulty Level:
Low
Properties of Determinants
Accuracy: 83%
Total Attempted30
Total Correct25
Total Incorrect5
Difficulty Level:
Low
Area of a Triangle Using Determinants
Accuracy: 85%
Total Attempted27
Total Correct23
Total Incorrect4
Difficulty Level:
Low

Learning Objectives

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Learning Objectives

  • Understand the concept of determinants and their significance in linear algebra.
  • Identify the properties of determinants up to order three.
  • Calculate the determinant of a square matrix.
  • Apply determinants to find the area of a triangle given its vertices.
  • Use determinants to solve systems of linear equations.
  • Understand the concept of minors and cofactors in relation to determinants.
  • Define and calculate the adjoint and inverse of a matrix using determinants.

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Revision Notes & Summary

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Chapter 4: Determinants

4.1 Introduction

  • Study of matrices and algebra of matrices.
  • Representation of systems of linear equations in matrix form.
  • Uniqueness of solutions determined by the determinant:
    • If
      • a₁b₂ - a₂b₁ ≠ 0, then the system has a unique solution.
  • Applications of determinants in various fields.

4.2 Determinant

  • Definition: A number associated with a square matrix A = [aᵢⱼ] of order n.

4.3 Area of a Triangle

  • Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is given by:
    A=12∣∣x1y11x2y21x3y31∣∣A = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|
  • Remarks:
    • Area is positive, take absolute value.
    • Collinear points yield an area of zero.

4.4 Adjoint and Inverse of a Matrix

  • Adjoint of a matrix: Transpose of the matrix of cofactors.
  • Inverse of a matrix: A⁻¹ exists if |A| ≠ 0.

4.5 Applications of Determinants and Matrices

  • Used for solving systems of linear equations and checking consistency.
  • Consistent system: Solution exists.
  • Inconsistent system: No solution exists.

4.6 Properties of Determinants

  • Determinant properties include:
    • |A| = 0 for singular matrices.
    • |A| ≠ 0 for non-singular matrices.

Examples

  • Example 1: Evaluate determinants using properties.
  • Example 2: Solve systems of equations using matrix methods.

Exercises

  1. Evaluate determinants in Exercises 1 and 2.
  2. Find area of triangles with given vertices.
  3. Show collinearity of points using determinants.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

The determinant is non-zero.

B.

The determinant is zero.

C.

The determinant is positive.

D.

The determinant is negative.
Correct Answer: B

Solution:

If any two columns of a matrix are proportional, the determinant of the matrix is zero.

Chapter Concept:

Zero Determinant Conditions

A.

−a21(a12a33−a13a32)+a22(a11a33−a13a31)−a23(a11a32−a12a31)-a_{21}(a_{12}a_{33} - a_{13}a_{32}) + a_{22}(a_{11}a_{33} - a_{13}a_{31}) - a_{23}(a_{11}a_{32} - a_{12}a_{31})

B.

a21(a13a32−a12a33)+a22(a11a33−a13a31)+a23(a12a31−a11a32)a_{21}(a_{13}a_{32} - a_{12}a_{33}) + a_{22}(a_{11}a_{33} - a_{13}a_{31}) + a_{23}(a_{12}a_{31} - a_{11}a_{32})

C.

a21(a32a13−a33a12)−a22(a33a11−a31a13)+a23(a31a12−a32a11)a_{21}(a_{32}a_{13} - a_{33}a_{12}) - a_{22}(a_{33}a_{11} - a_{31}a_{13}) + a_{23}(a_{31}a_{12} - a_{32}a_{11})

D.

−a21(a33a12−a32a13)+a22(a33a11−a31a13)−a23(a32a11−a31a12)-a_{21}(a_{33}a_{12} - a_{32}a_{13}) + a_{22}(a_{33}a_{11} - a_{31}a_{13}) - a_{23}(a_{32}a_{11} - a_{31}a_{12})
Correct Answer: A

Solution:

When expanding along the second row, the determinant is given by −a21(a12a33−a13a32)+a22(a11a33−a13a31)−a23(a11a32−a12a31)-a_{21}(a_{12}a_{33} - a_{13}a_{32}) + a_{22}(a_{11}a_{33} - a_{13}a_{31}) - a_{23}(a_{11}a_{32} - a_{12}a_{31}).

Chapter Concept:

Determinant of Order Three

A.

5

B.

-5

C.

0

D.

10
Correct Answer: B

Solution:

Interchanging two rows changes the sign of the determinant, so ∣A∣|A| becomes −5-5.

Chapter Concept:

Effects of Interchanging Rows or Columns

A.

When the matrix is singular

B.

When the determinant of the matrix is zero

C.

When the matrix is non-singular

D.

When the matrix is symmetric
Correct Answer: C

Solution:

The inverse of a matrix exists if and only if the matrix is non-singular, meaning its determinant is not zero.

Chapter Concept:

Adjoint and Inverse of a Matrix

A.

[3−1−52]\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}

B.

[3512]\begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}

C.

[2513]\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}

D.

[3152]\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}
Correct Answer: A

Solution:

The adjoint of a 2x2 matrix A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} is given by [d−b−ca]\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.

Chapter Concept:

Determinant of a Matrix

A.

12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|

B.

12∣x1(y3−y2)+x2(y1−y3)+x3(y2−y1)∣\frac{1}{2} \left| x_1(y_3-y_2) + x_2(y_1-y_3) + x_3(y_2-y_1) \right|

C.

12∣x1(y2+y3)+x2(y3+y1)+x3(y1+y2)∣\frac{1}{2} \left| x_1(y_2+y_3) + x_2(y_3+y_1) + x_3(y_1+y_2) \right|

D.

12∣x1(y1−y2)+x2(y2−y3)+x3(y3−y1)∣\frac{1}{2} \left| x_1(y_1-y_2) + x_2(y_2-y_3) + x_3(y_3-y_1) \right|
Correct Answer: A

Solution:

The correct formula for the area of a triangle using determinants is 12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|.

Chapter Concept:

Area of a Triangle Using Determinants

A.

The adjoint of AA is the inverse of AA.

B.

The adjoint of AA is the transpose of the cofactor matrix of AA.

C.

The adjoint of AA is the determinant of AA.

D.

The adjoint of AA is the same as AA.
Correct Answer: B

Solution:

The adjoint of a matrix AA is defined as the transpose of the matrix of cofactors of AA.

Chapter Concept:

Applications of Determinants in Solving Linear Equations

A.

Matrix CC is invertible.

B.

Matrix CC is singular.

C.

Matrix CC has a unique inverse.

D.

Matrix CC is non-singular.
Correct Answer: B

Solution:

A matrix is singular if its determinant is zero, meaning it does not have an inverse.

Chapter Concept:

Cofactor Expansion Theorem

A.

120

B.

1

C.

0

D.

15
Correct Answer: A

Solution:

The determinant of a diagonal matrix is the product of its diagonal elements. Thus, ∣A∣=4×5×6=120.|A| = 4 \times 5 \times 6 = 120.

Chapter Concept:

Determinant of Order Three

A.

All elements in a row are zero.

B.

Two rows are identical.

C.

Two rows are proportional.

D.

All of the above.
Correct Answer: D

Solution:

A determinant is zero if any row or column is entirely zero, if two rows or columns are identical, or if two rows or columns are proportional.

Chapter Concept:

Zero Determinant Conditions

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True or False

Correct Answer: False

Solution:

A square matrix AA is considered singular if ∣A∣=0|A| = 0. Non-singular matrices have a non-zero determinant.

Chapter Concept :

Determinant of a Matrix

Correct Answer: True

Solution:

According to the cofactor expansion theorem, if elements of one row (or column) are multiplied with the cofactors of any other row (or column), their sum is zero.

Chapter Concept :

Cofactor Expansion Theorem

Correct Answer: True

Solution:

The inverse of a matrix AA exists if and only if the determinant ∣A∣≠0|A| \neq 0, which means AA is non-singular. The formula for the inverse is A−1=1∣A∣⋅adj(A)A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A), where adj(A)\text{adj}(A) is the adjoint of AA.

Chapter Concept :

Adjoint and Inverse of a Matrix

Correct Answer: True

Solution:

Expanding a determinant along any row or column gives the same value, as shown in the expansion methods for different rows and columns.

Chapter Concept :

Expansion of Determinants

Correct Answer: True

Solution:

The determinant of a 3x3 matrix can be expanded along any row or column, and the value remains consistent due to the properties of determinants.

Chapter Concept :

Determinant of a Matrix

Correct Answer: True

Solution:

The given formula is correct for calculating the area of a triangle using the coordinates of its vertices.

Chapter Concept :

Area of a Triangle Using Determinants

Correct Answer: True

Solution:

The property of determinants states that interchanging any two rows or columns results in a change of sign of the determinant. This is a fundamental property used to simplify expressions and detect zero determinants.

Chapter Concept :

Effects of Interchanging Rows or Columns

Correct Answer: True

Solution:

The cofactor AijA_{ij} of an element aija_{ij} is defined as Aij=(−1)i+jMijA_{ij} = (-1)^{i+j} M_{ij}, where MijM_{ij} is the minor of aija_{ij}. This definition ensures that the cofactor accounts for the position of the element in the matrix, introducing a sign change based on the sum of the indices.

Chapter Concept :

Minors and Cofactors

Correct Answer: True

Solution:

A matrix is non-singular if its determinant is non-zero. The inverse of a matrix AA exists if and only if AA is non-singular, which means ∣A∣≠0|A| \neq 0.

Chapter Concept :

Adjoint and Inverse of a Matrix

Correct Answer: True

Solution:

A non-zero determinant indicates that the matrix is non-singular, and thus the system of equations has a unique solution.

Chapter Concept :

Applications of Determinants in Solving Linear Equations

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