Grade 11 Maths Unit 4:4.4 Properties of Determinants Exercise 4.5, 4.6 , 4.7 & 4.8

Introduction

This tutorial provides a step-by-step guide on the properties of determinants, as discussed in the Grade 11 Maths Unit 4.4 video by Saquama. Understanding determinants is crucial for solving matrices and equations in higher mathematics. This guide will summarize the exercises 4.5, 4.6, 4.7, and 4.8, helping students grasp the concepts effectively.

Step 1: Understanding Determinants

• A determinant is a scalar value that can be computed from the elements of a square matrix.

• It provides important information about the matrix, including whether it is invertible.

• For a 2x2 matrix, the determinant can be calculated using the formula:

  |A| = ad - bc
  

  where the matrix A is represented as:

  A = | a b |
      | c d |
  

Step 2: Properties of Determinants

1. Determinant of Identity Matrix:
   • The determinant of the identity matrix is always 1.
2. Determinant of Zero Matrix:
   • The determinant of a zero matrix is always 0.
3. Row Operations:
   • Swapping two rows changes the sign of the determinant.
   • Multiplying a row by a scalar multiplies the determinant by that scalar.
   • Adding a multiple of one row to another row does not change the determinant.

Step 3: Exercises Overview

Exercise 4.5

• Task: Calculate the determinant of a given 2x2 matrix.

• Example: For the matrix:

  | 2 3 |
  | 1 4 |
  

  Calculate the determinant:

  |A| = (2)(4) - (3)(1) = 8 - 3 = 5
  

Exercise 4.6

• Task: Demonstrate the effect of row operations on the determinant.
• Tip: Perform a row swap and observe the sign change in the determinant.

Exercise 4.7

• Task: Find the determinant of a 3x3 matrix.

• Formula: Use the rule of Sarrus or cofactor expansion.

  For a matrix:

  | a b c |
  | d e f |
  | g h i |
  

  The determinant is calculated as:

  |A| = a(ei - fh) - b(di - fg) + c(dh - eg)
  

Exercise 4.8

• Task: Apply the properties of determinants to solve a problem involving matrices.
• Practice: Choose a matrix, apply the properties discussed, and compute the determinant.

Conclusion

In this tutorial, we covered the fundamental properties of determinants and provided examples through exercises 4.5, 4.6, 4.7, and 4.8. Mastering these concepts is essential for progressing in matrix algebra and solving complex problems in mathematics. To deepen your understanding, practice these exercises and revisit the video for additional insights.