Grade 11 Math's Unit 4 Determinant and Their Properties part 1

Introduction

This tutorial provides a comprehensive guide to understanding determinants and their properties, specifically tailored for Grade 11 mathematics. It aims to break down complex concepts into manageable steps, making it easier to grasp the fundamentals of determinants, which are crucial in linear algebra and various applications in mathematics.

Step 1: Understanding Determinants

• Definition: 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.
• Notation: For a matrix A, the determinant is denoted as det(A) or |A|.

Step 2: Calculating Determinants of 2x2 Matrices

• Formula: For a 2x2 matrix

  A = | a  b |
      | c  d |
  

  The determinant is calculated using the formula:

  det(A) = ad - bc
  

• Example: For the matrix

  | 3  4 |
  | 2  5 |
  

  Calculate:

  det(A) = (3 * 5) - (4 * 2) = 15 - 8 = 7
  

Step 3: Calculating Determinants of 3x3 Matrices

• Formula: For a 3x3 matrix

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

  The determinant can be calculated using the rule of Sarrus or cofactor expansion:

  Rule of Sarrus:

  det(A) = aei + bfg + cdh - ceg - bdi - afh
  

• Example: For the matrix

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

  Calculate:

  det(A) = (1*5*9 + 2*6*7 + 3*4*8) - (3*5*7 + 2*4*9 + 1*6*8)
          = (45 + 84 + 96) - (105 + 72 + 48)
          = 225 - 225 = 0
  

Step 4: Properties of Determinants

• Property 1: The determinant of the identity matrix is 1.
• Property 2: If two rows (or columns) of a matrix are identical, the determinant is 0.
• Property 3: Swapping two rows (or columns) changes the sign of the determinant.
• Property 4: Multiplying a row (or column) by a scalar multiplies the determinant by the same scalar.

Step 5: Applications of Determinants

• Solving Linear Equations: Determinants can be used in Cramer's Rule to find solutions for systems of linear equations.
• Geometric Interpretation: The absolute value of a determinant represents the volume of the parallelepiped formed by the column vectors of the matrix.

Conclusion

This tutorial has outlined the foundational concepts of determinants, covering how to calculate them for 2x2 and 3x3 matrices, as well as their properties and applications. Understanding these concepts is crucial for progressing in mathematics, particularly in linear algebra. For further learning, practice calculating determinants of larger matrices and explore their applications in different mathematical contexts.