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To solve the problem, we need to find the value of \( k \) that makes the determinant of the given \( 2 \times 2 \) matrix equal to 5. The matrix is:

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

The determinant of a \( 2 \times 2 \) matrix is given by the formula:

\[
\text{determinant} = (a \cdot d) - (b \cdot c)
\]

For this matrix:

\[
\text{determinant} = (3 \cdot 5) - (k \cdot 1)
\]

Simplifying the equation:

\[
\text{determinant} = 15 - k
\]

We are told that the determinant is equal to 5, so:

\[
15 - k = 5
\]

Now, solve for \( k \):

\[
15 - 5 = k
\]

\[
k = 10
\]

Thus, the value of \( k \) that makes the determinant equal to 5 is \( k = 10 \).

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Would you like a more detailed explanation of the steps, or have any other questions?

Here are 5 related questions to further your understanding:
1. What happens to the determinant if \( k = 0 \) in this matrix?
2. Can you find the inverse of this matrix when \( k = 10 \)?
3. How would the result change if the matrix was \( 3 \times 3 \)?
4. What are the properties of a matrix with a determinant of 0?
5. How do we find the determinant of a matrix larger than \( 2 \times 2 \)?

**Tip**: In a \( 2 \times 2 \) matrix, swapping two rows or two columns changes the sign of the determinant.

El valor del K que hace que el determinante de la matriz sea 5:

Learn how to solve a linear algebra problem where you need to find the value of K that makes the determinant of a 2x2 matrix equal to 5. This example covers key concepts in matrix determinants and is suitable for high school students.

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