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Class 12 Science (PCM)

Mathematics Part 1

Matrices

Matrices

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Inverse Matrix Properties
Accuracy: 45%
Total Attempted22
Total Correct10
Total Incorrect12
Difficulty Level:
High
Matrix Multiplication
Accuracy: 47%
Total Attempted57
Total Correct27
Total Incorrect30
Difficulty Level:
High
Non-Commutativity of Matrix Multiplication
Accuracy: 50%
Total Attempted10
Total Correct5
Total Incorrect5
Difficulty Level:
Medium
Properties of Matrix Multiplication
Accuracy: 50%
Total Attempted14
Total Correct7
Total Incorrect7
Difficulty Level:
Medium
Zero Product Property Failure in Matrices
Accuracy: 50%
Total Attempted14
Total Correct7
Total Incorrect7
Difficulty Level:
Medium
Properties of Matrix Operations
Accuracy: 53%
Total Attempted38
Total Correct20
Total Incorrect18
Difficulty Level:
Medium
Conditions and Order of Matrix Products
Accuracy: 55%
Total Attempted67
Total Correct37
Total Incorrect30
Difficulty Level:
Medium
Properties of Matrix Addition and Scalar Multiplication
Accuracy: 56%
Total Attempted55
Total Correct31
Total Incorrect24
Difficulty Level:
Medium
Applications of Matrix Multiplication
Accuracy: 58%
Total Attempted12
Total Correct7
Total Incorrect5
Difficulty Level:
Medium
Symmetric and Skew-Symmetric Matrices
Accuracy: 60%
Total Attempted77
Total Correct46
Total Incorrect31
Difficulty Level:
Medium
Transpose Properties and Product Rule
Accuracy: 62%
Total Attempted13
Total Correct8
Total Incorrect5
Difficulty Level:
Medium
Matrix Equations and Systems
Accuracy: 64%
Total Attempted33
Total Correct21
Total Incorrect12
Difficulty Level:
Medium
Matrix Algebra
Accuracy: 67%
Total Attempted102
Total Correct68
Total Incorrect34
Difficulty Level:
Medium
Types of Matrices
Accuracy: 68%
Total Attempted56
Total Correct38
Total Incorrect18
Difficulty Level:
Medium
Equality of Matrices
Accuracy: 74%
Total Attempted46
Total Correct34
Total Incorrect12
Difficulty Level:
Medium
Invertible Matrices
Accuracy: 79%
Total Attempted56
Total Correct44
Total Incorrect12
Difficulty Level:
Low
Transpose of a Matrix
Accuracy: 83%
Total Attempted63
Total Correct52
Total Incorrect11
Difficulty Level:
Low
Construction of Matrices from General Term
Accuracy: 85%
Total Attempted39
Total Correct33
Total Incorrect6
Difficulty Level:
Low
Scalar Multiplication and Negative of Matrix
Accuracy: 90%
Total Attempted67
Total Correct60
Total Incorrect7
Difficulty Level:
Low
Matrix Definition, Elements and Order
Accuracy: 91%
Total Attempted54
Total Correct49
Total Incorrect5
Difficulty Level:
Low
Addition and Subtraction of Matrices
Accuracy: 93%
Total Attempted40
Total Correct37
Total Incorrect3
Difficulty Level:
Low
Decomposition into Symmetric and Skew-Symmetric Parts
Accuracy: 100%
Total Attempted17
Total Correct17
Total Incorrect0
Difficulty Level:
Low

Learning Objectives

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  • Understand the concept and definition of matrices.
  • Identify the various types of matrices, including symmetric and skew symmetric matrices.
  • Perform matrix operations such as addition, subtraction, and multiplication.
  • Apply the properties of matrix operations, including commutative, associative, and distributive laws.
  • Use matrices to solve systems of linear equations.
  • Explore the applications of matrices in different fields such as business, science, and cryptography.
  • Verify matrix identities using examples, such as (AB)′=B′A′(AB)' = B'A'.
  • Calculate the inverse of a matrix and understand its uniqueness.
  • Express any square matrix as the sum of a symmetric and a skew symmetric matrix.
  • Prove properties of matrices using mathematical induction and other methods.

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Revision Notes & Summary

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Matrices

Introduction

  • Matrices are essential in various branches of mathematics.
  • They simplify solving systems of linear equations and are used in electronic spreadsheets for business and science applications.
  • Matrices represent physical operations like magnification, rotation, and reflection, and are used in cryptography, genetics, economics, sociology, psychology, and industrial management.

Matrix Basics

  • A matrix is an ordered rectangular array of numbers or functions.
  • A matrix with mm rows and nn columns is of order m×nm \times n.
  • Types of matrices:
    • Column Matrix: [aij]m×1[a_{ij}]_{m \times 1}
    • Row Matrix: [aij]1×n[a_{ij}]_{1 \times n}
    • Square Matrix: When m=nm = n.
    • Diagonal Matrix: aij=0a_{ij} = 0 when i≠ji \neq j.
    • Scalar Matrix: aij=0a_{ij} = 0 when i≠ji \neq j, aij=ka_{ij} = k when i=ji = j.
    • Identity Matrix: aij=1a_{ij} = 1 when i=ji = j, aij=0a_{ij} = 0 when i≠ji \neq j.
    • Zero Matrix: All elements are zero.

Matrix Operations

  • Addition: A+B=B+AA + B = B + A
  • Multiplication: If A=[aij]m×nA = [a_{ij}]_{m \times n} and B=[bjk]n×pB = [b_{jk}]_{n \times p}, then AB=C=[cik]m×pAB = C = [c_{ik}]_{m \times p}.
  • Transpose: A′=[aji]n×mA' = [a_{ji}]_{n \times m}
    • Properties:
      • (A′)′=A(A')' = A
      • (kA)′=kA′(kA)' = kA'
      • (A+B)′=A′+B′(A + B)' = A' + B'
      • (AB)′=B′A′(AB)' = B'A'

Special Matrices

  • Symmetric Matrix: A′=AA' = A
  • Skew Symmetric Matrix: A′=−AA' = -A
  • Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.

Invertible Matrices

  • A square matrix AA is invertible if there exists a matrix BB such that AB=BA=IAB = BA = I.
  • The inverse of a matrix, if it exists, is unique.
  • If AA and BB are invertible, then (AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}.

Examples

  • Example 1: If A=[2314]A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, find A′A', A−1A^{-1}, and verify AA−1=IAA^{-1} = I.
  • Example 2: Given matrices AA and BB, show (A+B)′=A′+B′(A + B)' = A' + B'.

Applications

  • Used in solving linear equations, business applications, physical transformations, and various scientific fields.

Exercises

  • Prove properties of transpose and inverse matrices.
  • Solve matrix equations and verify results using matrix operations.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

All diagonal elements of AA are zero.

B.

All elements of AA are zero.

C.

AA is equal to its transpose.

D.

The sum of AA and its transpose is a skew-symmetric matrix.
Correct Answer: A

Solution:

In a skew-symmetric matrix, all diagonal elements are zero because aii=−aiia_{ii} = -a_{ii} implies 2aii=02a_{ii} = 0.

Chapter Concept:

Symmetric and Skew-Symmetric Matrices

A.

Matrices AA and BB are not equal.

B.

Matrices AA and BB are equal.

C.

Matrices AA and BB have different orders.

D.

Matrices AA and BB have different elements.
Correct Answer: B

Solution:

Matrices AA and BB are equal because they have the same order and all corresponding elements are equal.

Chapter Concept:

Equality of Matrices

A.

[69−312]\begin{bmatrix} 6 & 9 \\ -3 & 12 \end{bmatrix}

B.

[26−112]\begin{bmatrix} 2 & 6 \\ -1 & 12 \end{bmatrix}

C.

[36−14]\begin{bmatrix} 3 & 6 \\ -1 & 4 \end{bmatrix}

D.

[63−14]\begin{bmatrix} 6 & 3 \\ -1 & 4 \end{bmatrix}
Correct Answer: A

Solution:

Scalar multiplication involves multiplying each element of the matrix by the scalar. Thus, kA=3×[23−14]=[69−312]kA = 3 \times \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix} = \begin{bmatrix} 6 & 9 \\ -3 & 12 \end{bmatrix}.

Chapter Concept:

Scalar Multiplication and Negative of Matrix

A.

Both AA and BB are symmetric.

B.

Both AA and BB are skew-symmetric.

C.

AA is skew-symmetric and BB is symmetric.

D.

Neither AA nor BB is symmetric or skew-symmetric.
Correct Answer: C

Solution:

Matrix AA is skew-symmetric because A′=−AA' = -A, and matrix BB is symmetric because B′=BB' = B.

Chapter Concept:

Matrix Multiplication

A.

D=[−1977−11044]D = \begin{bmatrix} -19 & 77 \\ -110 & 44 \end{bmatrix}

B.

D=[19−77110−44]D = \begin{bmatrix} 19 & -77 \\ 110 & -44 \end{bmatrix}

C.

D=[0000]D = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

D.

D=[1111]D = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}
Correct Answer: A

Solution:

To find DD, solve CD−AB=OCD - AB = O. By substituting the given matrices and solving the system of equations, we find D=[−1977−11044]D = \begin{bmatrix} -19 & 77 \\ -110 & 44 \end{bmatrix}.

Chapter Concept:

Invertible Matrices

A.

A=12(A+A′)+12(A−A′)A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')

B.

A=A′+AA = A' + A

C.

A=A−A′A = A - A'

D.

A=A+A′A = A + A'
Correct Answer: A

Solution:

The decomposition of a square matrix AA into symmetric and skew-symmetric parts is given by A=12(A+A′)+12(A−A′)A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A'), where the first term is symmetric and the second term is skew-symmetric.

Chapter Concept:

Decomposition into Symmetric and Skew-Symmetric Parts

A.

Matrix multiplication is commutative.

B.

Matrix multiplication is associative.

C.

Matrix multiplication is distributive over addition.

D.

Matrix multiplication is both commutative and associative.
Correct Answer: B

Solution:

Matrix multiplication is associative, meaning (AB)C=A(BC)(AB)C = A(BC), but it is not commutative, meaning AB≠BAAB \neq BA in general.

Chapter Concept:

Matrix Multiplication

A.

[−1−2−3−4]\begin{bmatrix} -1 & -2 \\ -3 & -4 \end{bmatrix}

B.

[1−23−4]\begin{bmatrix} 1 & -2 \\ 3 & -4 \end{bmatrix}

C.

[−12−34]\begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix}

D.

[1234]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
Correct Answer: A

Solution:

The negative of a matrix is obtained by multiplying each element by -1. Therefore, the negative of BB is −B=[−1−2−3−4]-B = \begin{bmatrix} -1 & -2 \\ -3 & -4 \end{bmatrix}.

Chapter Concept:

Scalar Multiplication and Negative of Matrix

A.

ABAB is defined and has order m×pm \times p.

B.

ABAB is not defined.

C.

ABAB is defined and has order n×mn \times m.

D.

ABAB is defined and has order p×np \times n.
Correct Answer: A

Solution:

The product ABAB is defined when the number of columns in AA is equal to the number of rows in BB. The resulting matrix ABAB will have the order m×pm \times p.

Chapter Concept:

Conditions and Order of Matrix Products

A.

[1344]\begin{bmatrix} 1 & 3 \\ 4 & 4 \end{bmatrix}

B.

[3346]\begin{bmatrix} 3 & 3 \\ 4 & 6 \end{bmatrix}

C.

[2345]\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}

D.

[1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
Correct Answer: A

Solution:

The difference of matrices is obtained by subtracting corresponding elements: A−B=[2−13−04−05−1]=[1344]A - B = \begin{bmatrix} 2-1 & 3-0 \\ 4-0 & 5-1 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 4 & 4 \end{bmatrix}.

Chapter Concept:

Addition and Subtraction of Matrices

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True or False

Correct Answer: True

Solution:

By definition, a square matrix AA is symmetric if A′=AA' = A, meaning the transpose of the matrix is equal to the matrix itself.

Chapter Concept :

Symmetric and Skew-Symmetric Matrices

Correct Answer: True

Solution:

By definition, a skew symmetric matrix AA satisfies A′=−AA' = -A, and for all ii, aii=0a_{ii} = 0 because aii=−aiia_{ii} = -a_{ii} implies 2aii=02a_{ii} = 0.

Chapter Concept :

Types of Matrices

Correct Answer: True

Solution:

Matrix addition is defined as the element-wise addition of two matrices of the same order. If the matrices are not of the same order, addition is not defined.

Chapter Concept :

Addition and Subtraction of Matrices

Correct Answer: False

Solution:

Matrix multiplication is not commutative. Even if ABAB and BABA are both defined, it is not necessary that AB=BAAB = BA. This is a fundamental property of matrix multiplication.

Chapter Concept :

Properties of Matrix Operations

Correct Answer: True

Solution:

By definition, a matrix is an ordered rectangular array of numbers or functions. A matrix with mm rows and nn columns is of order m×nm \times n and contains mnmn elements.

Chapter Concept :

Matrix Definition, Elements and Order

Correct Answer: True

Solution:

Matrix addition is indeed commutative and associative for matrices of the same order, meaning that for any matrices AA and BB of the same order, A+B=B+AA + B = B + A and (A+B)+C=A+(B+C)(A + B) + C = A + (B + C). This property is fundamental to matrix operations.

Chapter Concept :

Properties of Matrix Addition and Scalar Multiplication

Correct Answer: True

Solution:

For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. This is a fundamental requirement for the multiplication to be defined.

Chapter Concept :

Matrix Equations and Systems

Correct Answer: True

Solution:

The inverse of a square matrix, if it exists, is unique. For invertible matrices AA and BB, (AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1} holds true.

Chapter Concept :

Invertible Matrices

Correct Answer: False

Solution:

The product ABAB is defined and has order m×pm \times p, but BABA is not necessarily defined unless m=pm = p. Therefore, both ABAB and BABA are defined only under specific order conditions.

Chapter Concept :

Conditions and Order of Matrix Products

Correct Answer: False

Solution:

For ABAB to be symmetric, it is necessary that AB=BAAB = BA. Symmetric matrices do not guarantee commutativity, so ABAB is not always symmetric unless AA and BB commute.

Chapter Concept :

Matrix Equations and Systems

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