Determinant of 3x3 Matrices, 2x2 Matrix, Precalculus Video Tutorial

Introduction

This tutorial will guide you through the process of finding the determinant of 2x2 and 3x3 matrices. Understanding determinants is crucial in precalculus and linear algebra, as they play a vital role in solving systems of equations, analyzing transformations, and more. By the end of this guide, you'll be equipped with the knowledge to calculate determinants confidently.

Step 1: Finding the Determinant of a 2x2 Matrix

To calculate the determinant of a 2x2 matrix, follow these steps:

1. Identify the Matrix:

   • A 2x2 matrix is structured as follows:

     | a  b |
     | c  d |
     

2. Apply the Determinant Formula:

   • Use the formula: [ \text{det}(A) = ad - bc ]
   • Here, a, b, c, and d are the elements of the matrix.

3. Calculate:

   • Multiply a and d.
   • Multiply b and c.
   • Subtract the second product from the first.

Example: For the matrix:

| 3  4 |
| 2  5 |

• Calculate: [ \text{det}(A) = (3)(5) - (4)(2) = 15 - 8 = 7 ]

Step 2: Finding the Determinant of a 3x3 Matrix

Calculating the determinant of a 3x3 matrix involves a more complex process. Here’s how to do it:

1. Identify the Matrix:

   • A 3x3 matrix looks like this:

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

2. Use the Determinant Formula:

   • The formula for the determinant is: [ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) ]

3. Calculate Using the Formula:

   • Break it down into the following steps:
     • Calculate ei - fh
     • Calculate di - fg
     • Calculate dh - eg
   • Substitute these values into the formula.

Example: For the matrix:

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

• Calculate:
  • ei - fh = (5)(9) - (6)(8) = 45 - 48 = -3
  • di - fg = (4)(9) - (6)(7) = 36 - 42 = -6
  • dh - eg = (4)(8) - (5)(7) = 32 - 35 = -3
• Substitute into the formula: [ \text{det}(A) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0 ]

Conclusion

You have now learned how to calculate the determinants of both 2x2 and 3x3 matrices. Remember, the determinant can provide insights into the properties of the matrix, such as whether it is invertible. Practice with different matrices to solidify your understanding, and consider exploring additional resources linked in the video description for further learning on matrices and precalculus.