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To find the value of \( k \) in the given matrix, we can use the property of determinants. Assuming that the determinant of the matrix is zero (which is often the case when solving for variables in such problems), we can calculate the determinant of the 4x4 matrix and solve for \( k \).

The matrix is:

\[
\begin{bmatrix}
2 & 1 & 3 & 5 \\
k & 2 & 1 & 3 \\
8 & k & 7 & 13 \\
8 & k & -3 & -1
\end{bmatrix}
\]

I will now compute the determinant and solve for \( k \).The determinant of the matrix is given by the expression:

\[
8k^2 - 64k + 128
\]

Solving this for \( k \) when the determinant is zero, we get:

\[
k = 4
\]

Therefore, the value of \( k \) is \( 4 \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the geometric interpretation of a determinant being zero?
2. Can we verify \( k = 4 \) by substituting it back into the matrix?
3. How does row or column operations affect the determinant?
4. What are the properties of determinants for larger matrices?
5. How would the determinant change if we adjusted other entries in the matrix?

**Tip**: When solving for variables in matrices, always check if there are conditions like zero determinant or specific properties that might simplify the solution process.

Discover how to calculate the determinant of a 4x4 matrix to find the value of k. This problem covers core linear algebra concepts and the condition for a singular matrix (determinant equals zero).

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