https://file.aiquickdraw.com/mathbot/file/1725475417941eip0bxvr.jpg

To find the determinant of the 3x3 matrix \( A \) given in the image:

\[
A = \begin{pmatrix} 
1 & 4 & 9 \\
7 & 2 & 5 \\
6 & 8 & 3 
\end{pmatrix}
\]

The determinant of a 3x3 matrix is calculated as:

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

For matrix \( A \):

\[
\text{det}(A) = 1 \cdot (2 \cdot 3 - 5 \cdot 8) - 4 \cdot (7 \cdot 3 - 5 \cdot 6) + 9 \cdot (7 \cdot 8 - 2 \cdot 6)
\]

Breaking it down:

- \( 2 \cdot 3 = 6 \)
- \( 5 \cdot 8 = 40 \)
- \( 7 \cdot 3 = 21 \)
- \( 5 \cdot 6 = 30 \)
- \( 7 \cdot 8 = 56 \)
- \( 2 \cdot 6 = 12 \)

So:

\[
\text{det}(A) = 1 \cdot (6 - 40) - 4 \cdot (21 - 30) + 9 \cdot (56 - 12)
\]

\[
\text{det}(A) = 1 \cdot (-34) - 4 \cdot (-9) + 9 \cdot 44
\]

\[
\text{det}(A) = -34 + 36 + 396
\]

\[
\text{det}(A) = 398
\]

Thus, the determinant of matrix \( A \) is \( \text{det}(A) = 398 \).

Would you like further details or have any questions? Here are some related questions you might consider:

1. How does the determinant help in understanding the properties of a matrix?
2. What is the geometric interpretation of a matrix determinant?
3. Can the determinant be used to determine if a matrix is invertible?
4. What is the relationship between the determinant and the eigenvalues of a matrix?
5. How does changing one element of the matrix affect the determinant?

**Tip:** When calculating the determinant of a matrix, it's often helpful to expand along a row or column with the most zeros to simplify calculations.

Learn how to calculate the determinant of a 3x3 matrix using the formula det(A) = a(ei - fh) - b(di - fg) + c(dh - eg). This problem involves concepts from linear algebra and is suitable for students in Grades 11-12.

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