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The image presents two matrices and asks to find their determinants using cofactor expansion.

The two matrices are:
1. 
\[
K_1 = \begin{bmatrix}
1 & 0 & 0 & 0 \\
3 & 6 & 0 & 0 \\
-1 & 5 & 7 & 0 \\
2 & -3 & 9 & -4
\end{bmatrix}
\]

2.
\[
K_2 = \begin{bmatrix}
-2 & 2 & 5 & 1 \\
0 & 1 & 4 & 8 \\
3 & 3 & 0 & 9 \\
1 & -5 & -2 & 0
\end{bmatrix}
\]

### Steps to compute the determinants using cofactor expansion:
1. **Matrix \( K_1 \):**
   - Since it has zeros in its first row (except for the first element), the cofactor expansion along the first row will simplify the calculation.
   
2. **Matrix \( K_2 \):**
   - Here, cofactor expansion can be applied along any row or column, but we will start with the first row to simplify the process.

Let me now compute the determinants for both matrices.The determinants of the two matrices are:

1. \(\text{det}(K_1) = -168\)
2. \(\text{det}(K_2) = -165\)

Would you like a detailed explanation of the cofactor expansion process, or have any other questions?

Here are 5 related questions to consider:
1. How does cofactor expansion work for 4x4 matrices?
2. Why is it beneficial to expand along a row or column with zeros?
3. Can you apply cofactor expansion for non-square matrices?
4. What other methods can be used to find determinants efficiently?
5. How do determinants relate to matrix invertibility?

**Tip:** When dealing with large matrices, it can be helpful to choose a row or column with the most zeros to simplify cofactor expansion.

Calculate the determinant of the following matrices using cofactor expansion:
1. Matrix K_1 = 
[1  0  0  0]
[3  6  0  0]
[-1  5  7  0]
[2  -3  9  -4]

2. Matrix K_2 = 
[-2  2  5  1]
[0  1  4  8]
[3  3  0  9]
[1  -5  -2  0]

Find the determinants of two 4x4 matrices using cofactor expansion. This problem covers key linear algebra concepts, including matrix determinants and the cofactor expansion theorem, with detailed step-by-step solutions.

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