To find the determinant of a \(3 \times 3\) matrix, you use the following method:

Given a matrix \( A \) as:

\[
A = \begin{pmatrix} 
a & b & c \\
d & e & f \\
g & h & i 
\end{pmatrix}
\]

The determinant of matrix \( A \), denoted as \( \text{det}(A) \), is calculated using:

\[
\text{det}(A) = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}
\]

Where each \( 2 \times 2 \) determinant is calculated as:

\[
\begin{vmatrix} e & f \\ h & i \end{vmatrix} = ei - fh
\]

So, expanding the determinant:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

This formula gives you the determinant of a \(3 \times 3\) matrix.

Let me know if you need further clarification or have more questions! Here are five questions related to this topic:
1. What is the determinant of a \(2 \times 2\) matrix?
2. How does the determinant of a matrix relate to the area or volume in geometry?
3. Can the determinant help in understanding the behavior of linear transformations?
4. What does a determinant of zero signify about a system of linear equations?
5. How is the determinant affected by row operations on a matrix?

**Tip:** The sign of the determinant can tell you if a linear transformation preserves orientation (positive determinant) or reverses it (negative determinant).

Learn how to calculate the determinant of a 3x3 matrix using the cofactor expansion method with a step-by-step explanation. This problem covers matrix operations and determinants, crucial for advanced high school students.

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