Determinants play a crucial role in linear algebra, with significant applications in various fields such as calculus, computer graphics, and physics. In this article, we will discuss how to calculate the determinant of a 3×3 matrix using a step-by-step approach.

Step 1: Understand a 3×3 matrix

A 3×3 matrix is a square grid containing nine cells arranged in three rows and three columns. Each cell contains an element, represented by the following notation:

| a b c |

| d e f |

| g h i |

The determinant of this matrix, denoted by |A| or det(A), is a scalar value that can be used to find various properties of the matrix.

Step 2: Apply the rule for calculating determinants

The determinant of a 3×3 matrix is calculated using this formula:

det(A) = (a(ei – fh) – b(di – fg) + c(dh – eg))

Step-by-step breakdown:

1. Calculate each product in the parentheses (you will have three sets):

Set 1: ei – fh

Set 2: di – fg

Set 3: dh – eg

2. Multiply each set by the corresponding factor from the first row:

Set 1: a(ei – fh)

Set 2: b(di – fg)

Set 3: c(dh – eg)

3. Add and subtract these products according to the formula:

det(A) = (a(ei – fh) – b(di – fg) + c(dh – eg))

Step 3: Example

Consider the following 3×3 matrix:

| 2 4 5 |

| 6 2 1 |

| 8 9 0 |

Using the determinant formula:

1. Calculate the products within parentheses:

Set 1: (2 * 0) – (9 * 1) = -9

Set 2: (6 * 1) – (2 * 0) = 6

Set 3: (8 * 5) – (4 * 1) = 36

2. Multiply each set by the corresponding factor from the first row:

Set 1: 2(-9) = -18

Set 2: 4(6) = 24

Set 3: 5(36) = 180

3. Apply the formula:

det(A) = (-18 – 24 + 180) = 138

Conclusion

Calculating the determinant of a 3×3 matrix is essential for various applications in mathematics and engineering. With practice, it becomes easier to find the determinant and use it for solving real-world problems.