https://file.aiquickdraw.com/mathbot/file/17271899116843guzzw22.jpg

The matrix in the image is:

\[
A = \begin{pmatrix}
2 & 3 & 1 \\
4 & 1 & 5 \\
3 & 2 & 0
\end{pmatrix}
\]

It seems like you are tasked with finding the determinant of this matrix using cofactor expansion or Sarrus' rule.

### To find the determinant using Sarrus' rule:
For a \(3 \times 3\) matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \), the determinant can be found as:

\[
\text{det}(A) = aei + bfg + cdh - ceg - bdi - afh
\]

### Let's calculate it for the matrix \( A \):

\[
A = \begin{pmatrix} 2 & 3 & 1 \\ 4 & 1 & 5 \\ 3 & 2 & 0 \end{pmatrix}
\]

Now apply the formula:

\[
\text{det}(A) = (2 \cdot 1 \cdot 0) + (3 \cdot 5 \cdot 3) + (1 \cdot 4 \cdot 2) - (1 \cdot 1 \cdot 3) - (3 \cdot 4 \cdot 0) - (2 \cdot 5 \cdot 2)
\]

Let me compute that for you.The determinant of the matrix \( A \) is:

\[
\text{det}(A) = 30
\]

Would you like further details or explanations on how the determinant was calculated?

Here are 5 related questions to expand your understanding:
1. What is cofactor expansion, and how does it compare to Sarrus' rule?
2. How would you compute the inverse of a \(3 \times 3\) matrix if the determinant is non-zero?
3. What are some practical applications of determinants in real-world problems?
4. How does Sarrus' rule apply to matrices larger than \(3 \times 3\)?
5. What is the significance of a matrix having a zero determinant?

**Tip:** A matrix with a zero determinant is called singular, meaning it doesn't have an inverse.

Find the determinant of matrix A using cofactor expansion or Sarrus' method. The matrix A is: 
 A = (2, 3, 1; 4, 1, 5; 3, 2, 0).

This page provides a detailed solution to finding the determinant of a 3x3 matrix using Sarrus' rule. The problem matrix is A = (2, 3, 1; 4, 1, 5; 3, 2, 0). Key concepts include matrix determinants, cofactor expansion, and Sarrus' rule.

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