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The problem asks to find the value of \( k \) that makes the determinant of the given \( 2 \times 2 \) matrix equal to zero.

The matrix is:

\[
\begin{pmatrix}
3 & k \\
1 & 5
\end{pmatrix}
\]

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as:

\[
\text{Determinant} = ad - bc
\]

For the given matrix, this becomes:

\[
\text{Determinant} = (3 \times 5) - (k \times 1) = 15 - k
\]

To make the determinant equal to zero, we set the equation:

\[
15 - k = 0
\]

Solving for \( k \):

\[
k = 15
\]

Thus, the value of \( k \) that makes the determinant zero is \( \boxed{15} \).

Would you like further explanation or details on how the determinant is calculated? 

Here are 5 related questions to deepen your understanding:
1. How do you calculate the determinant of a \( 3 \times 3 \) matrix?
2. What does it mean geometrically when the determinant of a matrix is zero?
3. How do determinants help in solving systems of linear equations?
4. What is the importance of the sign of a matrix's determinant?
5. How would the solution change if the matrix size increased?

**Tip:** In higher dimensions, a zero determinant indicates that the matrix is singular and non-invertible.

El valor del K que hace que el determinante de la matriz sea 0 es:

Discover how to find the value of k that makes the determinant of a 2x2 matrix equal to zero. The problem involves using the formula for the determinant of a 2x2 matrix and solving for k. Suitable for high school students in grades 9-12.

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