https://r2.erweima.ai/mathbot/image/1727100686829bzpd7xig.jpg

The task in the image is to calculate the determinant of a 5x5 matrix, denoted as \( D \). The matrix is:

\[
D = \begin{vmatrix}
1 & -1 & 2 & -3 & 1 \\
-3 & 3 & -7 & 9 & -5 \\
2 & 0 & 4 & -2 & 1 \\
3 & -5 & 7 & -14 & 6 \\
4 & -4 & 10 & -10 & 2 \\
\end{vmatrix}
\]

To solve this, I will compute the determinant of this 5x5 matrix using cofactor expansion.

Let me calculate the determinant for you.The determinant of the given 5x5 matrix is approximately \( D = 12 \).

Would you like to know more about how cofactor expansion works or have further questions?

Here are five related questions to explore:

1. How is cofactor expansion applied to compute the determinant of larger matrices like 5x5 or 4x4?
2. How does the determinant help in understanding the properties of a matrix, such as invertibility?
3. Can you explain how matrix row operations can simplify the calculation of determinants?
4. What is the geometric interpretation of a matrix's determinant?
5. How do determinants relate to solving systems of linear equations?

**Tip:** For larger matrices, using matrix decomposition techniques (e.g., LU decomposition) can simplify determinant calculations instead of expanding cofactors manually.

Calculate the determinant of the following 5x5 matrix: 
| 1  -1   2  -3   1 |
| -3  3  -7   9  -5 |
|  2   0   4  -2   1 |
|  3  -5   7 -14   6 |
|  4  -4  10 -10   2 |

This detailed solution explains how to calculate the determinant of a 5x5 matrix using cofactor expansion. The matrix is |1 -1 2 -3 1| | -3 3 -7 9 -5| | 2 0 4 -2 1| |3 -5 7 -14 6| |4 -4 10 -10 2|. Learn key concepts of matrix operations and determinants, suitable for advanced high school or college-level students.

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