Lesson Plan Math Class 12 (Ch-3) | Matrices

E- LESSON PLAN   SUBJECT MATHEMATICS    CLASS 10+2
Lesson Plan Class 12th chapter 3 Matrices Subject Mathematics for Mathematics Teacher. Effective way of Teaching Mathematics. Top planning by the teacher for effective teaching in the class.

TEACHER  :   DINESH KUMAR	SCHOOL :  RMB DAV CENTENARY PUBLIC SCHOOL NAWANSHAHR
SUBJECT   :   MATHEMATICS	CLASS                  :   XII  STANDARD BOARD                 :  CBSE
LESSON TOPIC / TITLE   :  CHAPTER 3: Matrices	ESTIMATED DURATION:  This lesson is divided into seven modules and it is completed in seven class meetings.

TOPIC:- Chapter : 3 : Matrices

DURATION:-  

This chapter is divided into eight modules and is completed in 15 class meetings.

PRE- REQUISITE KNOWLEDGE:-

Basic knowledge of algebra, and some simple functions.

TEACHING AIDS:- 

Green Board, Chalk,  Duster, Charts, smart board, projector, laptop etc.

METHODOLOGY:-   Lecture method

LEARNING OBJECTIVES:-

• Introduction and concepts of matrices with different definitions, addition, subtraction, multiplication of two matrices and multiplication of matrices with a scalar.

• Transpose of the matrices.

• Symmetric and skew symmetric matrices.

• Concept of elementary row and column operations.

• Method of finding the inverse of the matrices if exist by using elementary operations.

LEARNING  OUTCOMES:-

After studying this lesson students should know 

• different types of matrices, their addition, subtraction and multiplication. 
• Students should know the transpose of the matrix and are able to write the given matrix as the sum of symmetric and skew symmetric matrices. 
• Students should be able to find the inverse of matrices by applying elementary row or column operations.

RESOURCES

NCERT Text Book, 

NCERT Exemplar Book of mathematics, 

Resource Material : Worksheets , E-content, Basics and formulas from cbsemathematics.com)

KEY WORDS
Cartesian Product, Domain, Range, Reflexivity, Symmetry , Transitivity, One-One, Onto, Injective, Surjective , Bijective, composition

CONTENT OF THE TOPIC

1) Definition and terms associated with matrix.

2) Different types of matrices.

3) Operations on Matrices.

4) Symmeric and Skew Symmetric Matrices.

PROCEDURE

Start the session by asking some questions about the previous knowledge of the students and by introducing the concept of matrices in general form. Now introduce the different topics step by step as follows.

Matrices

Introduction of matrices, order of the matrices, elements of matrices, definitions of zero, identity, square and diagonal matrices.

Operations on Matrices

Operations on the matrices addition, subtraction, multiplication and multiplication of matrices with scalar.

Transpose of a matrix

If A is matrix then transpose of a matrix is obtained by interchanging its rows and columns and is denoted by A'

Properties of the transpose of the matrices

i)   (A')'  =  A,   or   Transpose of the transpose matrix is the matrix itself

ii)    (kA)'  =  kA',  where k is any scalar

iii)   (A + B)' = A' + B'

Orthogonal Matrices
Let A be a matrix and A' is its transpose.
A square matrix of order n is said to be orthogonal if its transpose is equal to its inverse. A' = A^-1

or

A square matrix of order n is said to be orthogonal if product of the matrix and its transpose gives a unit matrix
Mathematically: AA' = I

Symmetric and Skew symmetric matrices

Symmetric Matrix : A matrix A is said to be symmetric if  A' = A

Skew symmetric matrix : A matrix is said to be skew symmetric if A' = -A

Theorem 1:
For a square matrix A with real entries A + A' is always symmetric matrix and A - A' is always skew symmetric matrix.

Proof:   Let A + A' = B,  then

B' = (A + A')' = A' + (A')' = A' + A = A + A' = B

⇒ B = A + A' is a symmetric matrix

Let C = A - A' ,  then 

C' = (A - A')' = A' - (A')' = A' - A = - (A - A') = - C

Symmetric and Skew Symmetric Matrices

Transpose of the matrices and its properties, Symmetric and skew symmetric matrices.

Representation of square matrices as the sum of symmetric and skew symmetric matrix.

Existence of non-zero matrices (square matrices of order 2) whose product is zero.

Invertible matrices and proof of uniqueness of inverse, if it exists.

Properties of Skew symmetric matrices

All diagonal elements of a skew-symmetric matrix are zero.

For skew-symmetric matrices of even order (i.e., n × n with n even), the determinant can be non-zero.

If n is odd, the determinant of any n × n skew-symmetric matrix is zero.

The trace of a skew-symmetric matrix is zero.

Explanation: Since the trace is the sum of the diagonal elements and all diagonal elements of A are zero, the trace must also be zero.

Corresponding Elements on the either side of the diagonal are equal but with opposite sign.

 Invertible matrices

If A and B are two square matrices of the same order such that AB =BA = I, then B is said to be inverse of A and is denoted by A^-1  such that A^-1  = B. In this case A is said to be invertible.

In other words :

 If A is invertible then A^-1 exists  or If  A^-1 exists then A is said to be invertible.

Theorem 3 : Inverse of a matrix if exist then it is unique.

Proof:  
If possible let us suppose that the matrix A has two inverse B and C

B is the inverse of A then  AB = BA = I………… (i)

C is the inverse of A then  AC = CA = I ………(ii) 

From  (i) and (ii) we get

AB = AC  ⇒ B = C               ………….(Cancelling A on both side)

     ⇒ Both the inverse are same.

REFLECTION OF ACTIVITY

Students will be able to know the

• Different types of matrices.
• Addition, Subtraction and multiplication of matrices.
• Transpose of matrices.
• Symmetric and skew symmetric matrices.

IMMEDIATE FEEDBACK

After completing the above activities students will be able to 

1) Identify different types of matrices.

2) Able to perfom addition, subtraction and multiplication of two matrices.

3) Able to understand the the transpose, symmetric and skew symmetric matrices.

SUBJECTS INTEGRATED

Subject	Integration Concept	Example Using Matrices
Physics	Vectors & Transformations	Representing forces as matrices, solving systems of equations for circuit analysis, or applying rotation matrices in motion.
Computer Science	Data structures, image processing	Matrices are used to represent images (pixels as matrix elements), data in 2D arrays, and in algorithms for search/sort.
Economics	Input-output models, budgeting	Representing consumption and production data of industries, or calculating expenditures using matrix multiplication.
Chemistry	Reaction balancing, periodic data	Represent coefficients of chemical equations as matrices to balance reactions algebraically.
Art / Design	Transformations in 2D space	Use transformation matrices for scaling, rotation, or reflection in graphic design or architecture.
Physical Education	Performance tracking	Track fitness data (e.g., number of sit-ups, push-ups, laps) over a week for a group of students.

DIFFERENTIAL LEARNING

For Below Average Students

Mind/ Concept maps

For Average Students

• Learning situations through watching video, creating collage, completing puzzles, assignment (click here)

For Above Average Students:

Group Discussion

SKILLS ENHANCED

Observation  skill,  analytical skill,  critical thinking, team work, constructive approach, interpersonal skill,  engagement  in learning process etc.

Students will be given a Home Assignment for solving at home. 

• Assignment sheet will be given as home work at the end of the topic. 
• Separate sheets which will include questions of logical thinking and Higher order thinking skills will be given to the above average students.
• Class Test , Oral Test , worksheet and Assignments. can be made the part of assessment.
• Re-test(s) will be conducted on the basis of the performance of the students in the test.

Competency based assessment can be taken so as to ensure if the learning outcomes have been achieved or not. e.g.

• Puzzle
• Quiz
• Misconception check
• Peer check
• Students discussion
• Competency Based Assessment link: M C Q

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