Determinant Of 3x3 Matrix Worksheet With Examples

Matrices are a fundamental concept in linear algebra, and finding the determinant of a matrix is a crucial operation in many mathematical and real-world applications. In this article, we will explore the concept of determinants, specifically for 3x3 matrices, and provide a comprehensive worksheet with examples to help you practice and reinforce your understanding.

What is a Determinant?

The determinant of a matrix is a scalar value that represents certain properties of the matrix. It can be thought of as a measure of the "volume" of the parallelepiped spanned by the column vectors of the matrix. The determinant is denoted as det(A) or |A|, where A is the matrix.

Determinant of a 3x3 Matrix

To calculate the determinant of a 3x3 matrix, we can use the following formula:

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

where a, b, c, d, e, f, g, h, and i are the elements of the matrix A.

Step-by-Step Process

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

1. Write down the matrix and identify the elements.
2. Multiply the elements of the first row by the corresponding elements of the second and third rows.
3. Calculate the sum of the products from step 2.
4. Repeat steps 2 and 3 for the second and third rows.
5. Apply the formula det(A) = a(ei − fh) − b(di − fg) + c(dh − eg).

Worksheet with Examples

Here are some examples to help you practice finding the determinant of a 3x3 matrix:

Example 1

Find the determinant of the matrix:

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

Solution

Using the formula, we get:

det(A) = 1(5(9) − 6(8)) − 2(4(9) − 6(7)) + 3(4(8) − 5(7)) = 1(45 − 48) − 2(36 − 42) + 3(32 − 35) = 1(-3) − 2(-6) + 3(-3) = -3 + 12 − 9 = 0

Example 2

Find the determinant of the matrix:

| 2  1  4 |
| 3  5  2 |
| 1  3  6 |

Solution

Using the formula, we get:

det(A) = 2(5(6) − 2(3)) − 1(3(6) − 2(1)) + 4(3(3) − 5(1)) = 2(30 − 6) − 1(18 − 2) + 4(9 − 5) = 2(24) − 1(16) + 4(4) = 48 − 16 + 16 = 48

Example 3

Find the determinant of the matrix:

| 1  0  2 |
| 3  4  1 |
| 2  5  3 |

Solution

Using the formula, we get:

det(A) = 1(4(3) − 1(5)) − 0(3(3) − 1(2)) + 2(3(5) − 4(2)) = 1(12 − 5) − 0(9 − 2) + 2(15 − 8) = 1(7) − 0(7) + 2(7) = 7 + 14 = 21

Gallery of Determinant Examples

Frequently Asked Questions

We hope this worksheet and examples have helped you understand the concept of determinants for 3x3 matrices. Practice makes perfect, so be sure to try out the examples and exercises to reinforce your understanding.

Gallery of Determinant Of 3x3 Matrix Worksheet With Examples