Matrices I

Grade 12 · Mathematics

Semester 1 | Period 1 | Week 5

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Subject: Mathematics

Semester: 1

Period: 1

Week: 5

WEEK 5

Class: Grade 12
Age: 17 years
Duration: 40 minutes per period, 5 periods
Subject: Mathematics
Topic: Matrices I
Focus: Definition of matrices, order and notation of matrices, types of matrices, addition and subtraction of matrices, scalar multiplication of 2x2 and 3x3 matrices.

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

  1. Define a matrix and understand its components.
  2. Understand the order and notation of matrices.
  3. Identify different types of matrices such as null, unit, and square matrices.
  4. Perform addition and subtraction of matrices.
  5. Perform scalar multiplication of 2x2 and 3x3 matrices.

 

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Discussion
  • Practice exercises
  • Analogies and real-life connections

INSTRUCTIONAL MATERIALS:

  • Matrix charts
  • Matrix addition and subtraction charts
  • Determinant charts
  • Computer-assorted instructional materials
  • Whiteboard and markers

 

PERIOD 1: Introduction to Matrices

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of a matrix, its components (rows and columns). Explains that a matrix is a rectangular array of numbers.

Students listen attentively and ask questions.

Step 2 - Matrix Notation

Explains matrix notation (A = [a₁₁ a₁₂; a₂₁ a₂₂]) and its order (m x n). Demonstrates writing matrices using proper notation.

Students observe and take notes on matrix notation.

Step 3 - Matrix Types

Introduces different types of matrices: null (zero) matrix, unit (identity) matrix, square matrix, and row/column matrices.

Students observe examples of different types and identify them.

Step 4 - Practice

Leads students through examples of matrix types and notation.

Students practice identifying matrices in different forms.

NOTE ON BOARD:

  • A matrix is an ordered rectangular array of numbers.
  • Notation: A = [a₁₁ a₁₂; a₂₁ a₂₂]
  • Types:
    • Null Matrix: A matrix with all zero elements.
    • Unit Matrix: A square matrix with ones on the diagonal and zeros elsewhere.
    • Square Matrix: A matrix with the same number of rows and columns.
    • Row Matrix: A matrix with only one row.
    • Column Matrix: A matrix with only one column.

EVALUATION:

  1. What is a matrix?
  2. Write the matrix notation for a 2x2 matrix.
  3. Identify the type of matrix: [1 0; 0 1].
  4. How do you denote a square matrix?
  5. Explain the difference between a row matrix and a column matrix.

CLASSWORK:

  1. Identify the matrix type: [0 0; 0 0].
  2. Write a 3x2 matrix.
  3. Identify the type of matrix: [3 4 5].
  4. What is the order of matrix B = [a b; c d; e f]?
  5. Define a null matrix.

ASSIGNMENT:

  1. Find the order of matrix A = [1 2 3; 4 5 6].
  2. Explain the importance of the unit matrix.
  3. Write a 3x3 unit matrix.
  4. Identify and write a row matrix.
  5. Research the use of matrices in computer graphics.

 

PERIOD 2 & 3: Addition and Subtraction of Matrices

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Explains how matrix addition and subtraction work, emphasizing that the matrices must have the same order.

Students listen and take notes.

Step 2 - Matrix Addition

Demonstrates matrix addition by adding two 2x2 matrices. Example: A = [1 2; 3 4], B = [5 6; 7 8], A + B = [6 8; 10 12].

Students observe and write down the procedure.

Step 3 - Matrix Subtraction

Demonstrates matrix subtraction by subtracting two 2x2 matrices. Example: A = [5 6; 7 8], B = [1 2; 3 4], A - B = [4 4; 4 4].

Students observe and write down the steps for subtraction.

Step 4 - Guided Practice

Leads students through several examples of addition and subtraction of matrices.

Students practice in pairs, discussing the results.

NOTE ON BOARD:

  • Matrix Addition: If A = [a₁₁ a₁₂; a₂₁ a₂₂] and B = [b₁₁ b₁₂; b₂₁ b₂₂], then A + B = [a₁₁ + b₁₁ a₁₂ + b₁₂; a₂₁ + b₂₁ a₂₂ + b₂₂].
  • Matrix Subtraction: If A = [a₁₁ a₁₂; a₂₁ a₂₂] and B = [b₁₁ b₁₂; b₂₁ b₂₂], then A - B = [a₁₁ - b₁₁ a₁₂ - b₁₂; a₂₁ - b₂₁ a₂₂ - b₂₂].

EVALUATION:

  1. Perform the addition of two 3x3 matrices.
  2. Subtract the following matrices: [5 6; 7 8] - [1 2; 3 4].
  3. What is the condition for adding or subtracting matrices?
  4. Add the following matrices: [1 1; 1 1] and [2 2; 2 2].
  5. Subtract the following matrices: [8 9; 10 11] - [4 5; 6 7].

CLASSWORK:

  1. Add the matrices: [2 3; 4 5] and [1 0; 6 7].
  2. Subtract the matrices: [10 12; 14 16] and [2 3; 4 5].
  3. Add two matrices: [5 5; 5 5] and [0 0; 0 0].
  4. Subtract: [7 8; 9 10] - [1 2; 3 4].
  5. Perform matrix addition for [1 2] and [3 4].

ASSIGNMENT:

  1. Add the matrices: [1 3; 4 2] and [2 5; 6 8].
  2. Subtract the matrices: [6 7; 8 9] - [2 1; 4 3].
  3. Provide the conditions for matrix addition.
  4. Solve: [4 5; 6 7] + [3 2; 1 8].
  5. Research the use of matrix operations in solving real-world problems.

 

PERIOD 4 & 5: Scalar Multiplication of 2x2 and 3x3 Matrices

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Explains scalar multiplication, where each element of the matrix is multiplied by a constant (scalar).

Students listen attentively.

Step 2 - Scalar Multiplication of 2x2 Matrices

Demonstrates scalar multiplication of a 2x2 matrix. Example: Multiply matrix A = [1 2; 3 4] by 2: 2A = [2 4; 6 8].

Students observe the steps and take notes.

Step 3 - Scalar Multiplication of 3x3 Matrices

Demonstrates scalar multiplication of a 3x3 matrix. Example: Multiply matrix B = [1 2 3; 4 5 6; 7 8 9] by 3: 3B = [3 6 9; 12 15 18; 21 24 27].

Students observe and take notes on the procedure.

Step 4 - Guided Practice

Leads students through examples of scalar multiplication for both 2x2 and 3x3 matrices.

Students practice scalar multiplication in pairs.

NOTE ON BOARD:

  • Scalar multiplication: If A = [a₁₁ a₁₂; a₂₁ a₂₂], and k is a scalar, then kA = [ka₁₁ ka₁₂; ka₂₁ ka₂₂].
  • Scalar multiplication of 3x3: If B = [b₁₁ b₁₂ b₁₃; b₂₁ b₂₂ b₂₃; b₃₁ b₃₂ b₃₃] and k is a scalar, then kB = [kb₁₁ kb₁₂ kb₁₃; kb₂₁ kb₂₂ kb₂₃; kb₃₁ kb₃₂ kb₃₃].

EVALUATION:

  1. Multiply the matrix [1 2; 3 4] by 3.
  2. Multiply the matrix [2 4 6; 8 10 12; 14 16 18] by 2.
  3. Explain the concept of scalar multiplication.
  4. Multiply the matrix [0 1; 2 3] by 4.
  5. Multiply the matrix [3 6; 9 12] by 5.

CLASSWORK:

  1. Multiply the matrix [2 3; 4 5] by 2.
  2. Multiply the matrix [1 1 1; 2 2 2; 3 3 3] by 3.
    Scalar multiply the matrix [5 6; 7 8] by 2.
    4. Perform scalar multiplication on the matrix [1 3 5; 2 4 6; 7 8 9].
    5. Multiply [3 3; 5 5] by 4.

ASSIGNMENT:

  1. Multiply the matrix [1 2; 3 4] by 6.
  2. Scalar multiply the matrix [7 8 9; 10 11 12; 13 14 15] by 3.
  3. Solve: Multiply [4 6; 2 4] by 5.
  4. Find the result of multiplying [2 3 4; 5 6 7; 8 9 10] by 2.
  5. Explain how scalar multiplication helps in solving matrix equations.