Matrices, Definition, Types and Operations for SS 3

Class: Senior Secondary School 3 (SS3 / SSS 3)
Term: 1st Term
Week: 3
Age: 17 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra
Previous Lesson:
Topic: MATRICES – I
Subject Matter: Definition of matrix, order and notation of matrix, types of matrices (null, unit, etc.), addition and subtraction of matrices, scalar multiplication of 2×2 and 3×3 matrices

Specific Objectives

By the end of the lesson, students should be able to:

Cognitive Domain:

• Define a matrix.
• State the order and notation of a given matrix.
• Identify and describe different types of matrices.
• Perform addition and subtraction of matrices of the same order.
• Perform scalar multiplication on 2×2 and 3×3 matrices.

Affective Domain:

• Appreciate the systematic nature of matrix operations.
• Show interest in solving problems involving matrices.

Psychomotor Domain:

• Accurately write down matrices based on given order.
• Correctly carry out matrix addition, subtraction, and scalar multiplication.

Social Domain:

• Collaborate with peers to solve matrix problems.

Reference Materials

The following resources were used in planning this lesson:

• FCT-ERC SS 3 Scheme of Work, September 2014 curriculum.
• Relevant State Unified Scheme of Work.
• A suitable General Mathematics textbook for SS 3.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Matrix charts
• Matrix addition charts
• Subtraction charts
• Determinant charts
• Computer instructional materials

Rationale for the Lesson

This lesson helps students understand matrices, which are important for solving systems of linear equations and are used in various fields like computer graphics, engineering, and economics. Learning about matrices enables students to develop logical thinking and problem-solving skills that are useful in their academic pursuits and daily lives.

Prerequisite/Previous Knowledge

Students are expected to have a basic understanding of numbers, variables, and simple algebraic operations.

Lesson Content/Board Summary

MATRICES – I

Definition of a Matrix

A matrix is a rectangular array or arrangement of numbers, symbols, or expressions, called elements, arranged in rows and columns. Matrices are usually enclosed in square brackets [ ] or round brackets ( ).

Order and Notation of a Matrix

The order (or dimension) of a matrix describes its size, given by the number of rows by the number of columns (m x n).

1. If a matrix has ‘m’ rows and ‘n’ columns, its order is m x n.
2. Matrices are generally denoted by capital letters (e.g., A, B, C).
3. Elements within a matrix are denoted by lowercase letters with subscripts indicating their position (e.g., a_ij for the element in the i-th row and j-th column).

Example:

A = [ 2 -1 ]

[ 3 4 ]

This matrix A has 2 rows and 2 columns, so its order is 2×2.

Types of Matrices

Matrices can be classified into different types based on their structure and elements. The following are common types of matrices:

1. Row Matrix: A matrix with only one row.

   Example: B = [ 1 5 -2 ] (Order 1×3)

2. Column Matrix: A matrix with only one column.

   Example: C = [ 4 ]

   [ 0 ]

   [ 7 ] (Order 3×1)

3. Square Matrix: A matrix where the number of rows is equal to the number of columns (m = n).

   Example: D = [ 2 -1 ]

   [ 3 4 ] (Order 2×2)

4. Null Matrix (Zero Matrix): A matrix in which all its elements are zero. It is usually denoted by O.

   Example: O = [ 0 0 ]

   [ 0 0 ] (Order 2×2)

5. Unit Matrix (Identity Matrix): A square matrix in which all the elements on the main diagonal are 1, and all other elements are 0. It is usually denoted by I.

   Example: I = [ 1 0 ]

   [ 0 1 ] (Order 2×2)

6. Diagonal Matrix: A square matrix in which all the non-diagonal elements are zero.

   Example: F = [ 5 0 ]

   [ 0 2 ] (Order 2×2)

7. Scalar Matrix: A diagonal matrix where all the diagonal elements are equal.

   Example: G = [ 3 0 ]

   [ 0 3 ] (Order 2×2)

Addition and Subtraction of Matrices

Matrices can be added or subtracted only if they have the same order (i.e., the same number of rows and columns). The operation is performed by adding or subtracting corresponding elements.

Steps for Addition/Subtraction:

1. Check if the matrices have the same order. If not, addition/subtraction is not possible.
2. Add or subtract the elements in the corresponding positions.

Example 1 (Addition):

Given matrices A = [ 1 2 ] and B = [ 5 6 ]

[ 3 4 ] [ 7 8 ]

Find A + B.

Solution:

Step 1: Identify the order of A and B. Both are 2×2 matrices. Addition is possible.

Step 2: Add the corresponding elements.

A + B = [ (1+5) (2+6) ]

[ (3+7) (4+8) ]

A + B = [ 6 8 ]

[ 10 12 ]

Example 2 (Subtraction):

Given matrices P = [ 10 -3 ] and Q = [ 4 2 ]

[ 5 8 ] [ 1 -6 ]

Find P – Q.

Solution:

Step 1: Identify the order of P and Q. Both are 2×2 matrices. Subtraction is possible.

Step 2: Subtract the corresponding elements.

P – Q = [ (10-4) (-3-2) ]

[ (5-1) (8-(-6)) ]

P – Q = [ 6 -5 ]

[ 4 14 ]

Scalar Multiplication of Matrices

Scalar multiplication involves multiplying a matrix by a single number (scalar). Each element of the matrix is multiplied by the scalar.

Steps for Scalar Multiplication:

1. Take the scalar value.
2. Multiply every element inside the matrix by this scalar value.

Example 1 (2×2 Matrix):

Given scalar k = 3 and matrix A = [ 1 2 ]

[ 4 5 ]

Find kA.

Solution:

Step 1: Multiply each element of matrix A by the scalar k = 3.

kA = 3 * [ 1 2 ]

[ 4 5 ]

kA = [ (3*1) (3*2) ]

[ (3*4) (3*5) ]

kA = [ 3 6 ]

[ 12 15 ]

Example 2 (3×3 Matrix):

Given scalar m = -2 and matrix B = [ 1 0 -1 ]

[ 2 3 4 ]

[ 5 1 0 ]

Find mB.

Solution:

Step 1: Multiply each element of matrix B by the scalar m = -2.

mB = -2 * [ 1 0 -1 ]

[ 2 3 4 ]

[ 5 1 0 ]

mB = [ (-2*1) (-2*0) (-2*-1) ]

[ (-2*2) (-2*3) (-2*4) ]

[ (-2*5) (-2*1) (-2*0) ]

mB = [ -2 0 2 ]

[ -4 -6 -8 ]

[ -10 -2 0 ]

Quick Revision Table

Area	Key Point
Definition of a Matrix	A matrix is a rectangular array or arrangement of numbers, symbols, or expressions, called elements, arranged in rows and columns. Matrices are…
Order and Notation of a Matrix	The order (or dimension) of a matrix describes its size, given by the number of rows by the number of columns (m x n). If a matrix has 'm' rows…
Types of Matrices	Matrices can be classified into different types based on their structure and elements. The following are common types of matrices: Row Matrix: A…
Addition and Subtraction of Matrices	Matrices can be added or subtracted only if they have the same order (i.e., the same number of rows and columns). The operation is performed by…
Scalar Multiplication of Matrices	Scalar multiplication involves multiplying a matrix by a single number (scalar). Each element of the matrix is multiplied by the scalar. Steps for…

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction and Recall

Time: 4 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher greets the students and introduces the topic by asking them if they have ever seen data arranged in rows and columns, like in a timetable or a spreadsheet. The teacher then explains that such arrangements are formally called matrices in mathematics.

Students’ Activity: Students respond to the teacher's questions and listen attentively to the introduction.

Learning Point: Prior matrix knowledge.

Step 2: Definition of a Matrix

Time: 5 minutes

Teaching Skill: Explanation/Definition

Teacher’s Activity: The teacher guides students to define a matrix as a rectangular array of numbers or functions arranged in rows and columns. The teacher writes the definition and an example on the board.

Students’ Activity: Students participate in defining a matrix and copy the definition and example into their notebooks.

Learning Point: Meaning of matrix.

Step 3: Order and Notation of a Matrix

Time: 5 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains how to determine the order of a matrix (rows x columns) and demonstrates matrix notation using capital letters for the matrix and lowercase letters with subscripts for elements. An example matrix is given, and its order and element notation are identified.

Students’ Activity: Students identify the order of given matrices and practice writing elements using notation.

Learning Point: Matrix problem-solving methods.

Step 4: Types of Matrices

Time: 5 minutes

Teaching Skill: Classification/Explanation

Teacher’s Activity: The teacher identifies and explains various types of matrices, such as row, column, square, null, and unit/identity matrices, providing examples for each type. The teacher encourages students to ask questions for clarification.

Students’ Activity: Students listen, ask questions, and copy the different types of matrices and their examples.

Learning Point: Types of matrix.

Step 5: Addition and Subtraction of Matrices

Time: 5 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher explains the condition for adding or subtracting matrices (same order). The teacher demonstrates the process by solving examples of matrix addition and subtraction on the board, showing step-by-step calculations.

Students’ Activity: Students pay attention, ask questions, and practice solving similar problems in their notebooks.

Learning Point: Matrix problem-solving methods.

Step 6: Scalar Multiplication of Matrices

Time: 5 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher explains scalar multiplication, emphasizing that every element in the matrix is multiplied by the scalar. The teacher solves examples involving 2×2 and 3×3 matrices on the board.

Students’ Activity: Students observe the steps, ask questions, and work through the examples in their notebooks.

Learning Point: Matrix problem-solving methods.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define a matrix.
2. State the order of the matrix A = [ 1 0 -2 ]
3. Mention any three types of matrices.
4. Given A = [ 2 1 ] and B = [ 3 0 ], find A + B.
5. Given C = [ 4 -1 ] and scalar k = 2, find kC.

Students’ Activity: Students answer orally and in writing, showing working or giving examples where the lesson requires it.

Learning Point: Matrix skills assessed.

Step 8: Note-Taking

Time: 8 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides students to copy the essential Board Summary on matrix into their notebooks, drawing attention to the key rules, examples, diagrams or distinctions already taught.

Students’ Activity: Students copy the essential notes carefully into their notebooks and correct any errors in their earlier work.

Learning Point: Matrix notes recorded.

Step 9: Conclusion

Time: 3 minutes

Teaching Skill: Reinforcement/Summarization

Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the definition, order, types, and basic operations (addition, subtraction, scalar multiplication) of matrices. The teacher gives homework by asking students to find real-life examples where matrices are used.

Students’ Activity: Students listen to the summary.

Learning Point: Matrix concepts consolidated.

Continuous Assessment/Further Study

Type: Practice Exercise

Instruction: Complete the following tasks in your notebook. Show all mathematical working clearly.

1. For \(A=\begin{pmatrix}2&-1\\3&4\end{pmatrix}\), state the order of the matrix.
2. Given \(A=\begin{pmatrix}2&1\\3&5\end{pmatrix}\) and \(B=\begin{pmatrix}4&-2\\1&3\end{pmatrix}\), find \(A+B\).
3. Using the same matrices, find \(A-B\).
4. Find \(3A\) for \(A=\begin{pmatrix}1&-2\\4&0\end{pmatrix}\).
5. Give one example each of a zero matrix, identity matrix and diagonal matrix.

Lesson Keywords

• Matrix – A rectangular array of numbers or expressions arranged in rows and columns.
• Order – The dimension of a matrix, given by the number of rows by the number of columns (m x n).
• Scalar – A single number that multiplies every element of a matrix.
• Null Matrix – A matrix where all elements are zero.
• Unit Matrix – A square matrix with 1s on the main diagonal and 0s elsewhere.

Differentiation

For students who grasp concepts quickly, the teacher can provide additional complex problems involving a combination of operations. For students needing more support, the teacher can provide simpler matrices with fewer elements or offer one-on-one guidance during practice sessions.

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