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The image shows a matrix with the instruction in Greek, asking to calculate the determinant of the matrix. The matrix is:

\[
\begin{bmatrix}
5 & 2 & 3 & 2 & 0 & 0 \\
6 & 2 & 3 & 2 & 0 & 0 \\
5 & 3 & 3 & 2 & 0 & 0 \\
5 & 2 & 3 & 3 & 0 & 0 \\
2 & 1 & 8 & 3 & 4 & 5 \\
3 & 6 & 9 & 0 & 3 & 4
\end{bmatrix}
\]

Let's calculate the determinant of this 6x6 matrix.

I will proceed with the calculation now.The determinant of the given matrix is approximately \( 3 \).

Would you like further details on how the determinant was computed or explanations of each step?

Here are five follow-up questions based on this problem:

1. How is the determinant of a 6x6 matrix typically computed?
2. What does the determinant of a matrix represent in linear algebra?
3. How can you use row reduction to simplify the calculation of a determinant?
4. In what situations is the determinant of a matrix zero, and what does it signify?
5. How does the determinant relate to the invertibility of a matrix?

**Tip:** For larger matrices, using a numerical method or software is often more efficient than manual calculation, especially when the matrix contains complex values.

Calculate the determinant of the given 6x6 matrix.

Learn how to calculate the determinant of a 6x6 matrix with a detailed step-by-step solution. This guide covers key concepts in linear algebra, including the use of cofactor expansion and row reduction methods. Suitable for undergraduate students or advanced high school math learners.

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