Grade 11 Maths Unit 4:4.3 Determinants of Matrices of Order 3 & Exercise 4 3 and Exercise 4. 4

Introduction

This tutorial aims to guide you through the concepts of determinants of matrices of order 3, along with exercises that reinforce these concepts. Understanding determinants is crucial in linear algebra, as they provide insights into matrix properties, such as invertibility and volume scaling.

Step 1: Understanding Determinants

• Definition: A determinant is a scalar value that can be computed from the elements of a square matrix. It is denoted as det(A) or |A| for a matrix A.
• Importance: Determinants help in:
  • Determining if a matrix is invertible (a non-zero determinant indicates the matrix is invertible).
  • Solving systems of linear equations.
  • Calculating areas and volumes in geometry.

Step 2: Determinant of a 3x3 Matrix

To calculate the determinant of a 3x3 matrix, follow these steps:

1. Matrix Representation: Consider a 3x3 matrix:

   A = | a11 a12 a13 |
       | a21 a22 a23 |
       | a31 a32 a33 |
   

2. Use the Formula: The formula for the determinant of matrix A is:

   det(A) = a11 * (a22 * a33 - a23 * a32) 
          - a12 * (a21 * a33 - a23 * a31) 
          + a13 * (a21 * a32 - a22 * a31)
   

3. Calculate Step-by-Step:

   • Calculate the minors for each element in the first row.
   • Multiply each element by the determinant of the corresponding 2x2 submatrix.
   • Apply the signs according to the cofactor expansion (positive for a11, negative for a12, positive for a13).

Step 3: Example Calculation

Let’s calculate the determinant of the following matrix:

A = | 1 2 3 |
    | 0 4 5 |
    | 1 6 0 |

1. Apply the formula:

   det(A) = 1 * (4 * 0 - 5 * 6) 
          - 2 * (0 * 0 - 5 * 1) 
          + 3 * (0 * 6 - 4 * 1)
   

2. Simplify:

   • Calculate each term:
     • First term: 1 * (0 - 30) = -30
     • Second term: -2 * (0 - 5) = +10
     • Third term: 3 * (0 - 4) = -12
   • Combine: -30 + 10 - 12 = -32

   Therefore, det(A) = -32.

Step 4: Practice Exercises

To reinforce your understanding, complete the following exercises:

1. Find the determinant of the matrix:

   B = | 2 3 1 |
       | 4 5 6 |
       | 7 8 9 |
   

2. Solve and verify the determinant of:

   C = | 3 1 2 |
       | 1 3 1 |
       | 2 1 3 |
   

Conclusion

In this tutorial, you learned about the determinants of 3x3 matrices, how to calculate them using a specific formula, and practiced with example matrices. Understanding these concepts is essential for further studies in linear algebra. As a next step, consider exploring determinants of larger matrices or applying these concepts to solve systems of linear equations.