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To determine for which value of \( k \) the vectors \( \mathbf{v}_1 \), \( \mathbf{v}_2 \), and \( \mathbf{v}_3 \) are linearly dependent, we need to check when their determinant is zero. The vectors are:

\[
\mathbf{v}_1 = \begin{pmatrix} 4 \\ 0 \\ 2 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 0 \\ 2 \\ 4 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} k \\ 4 \\ 2 \end{pmatrix}
\]

For the vectors to be linearly dependent, the determinant of the matrix formed by these vectors must be zero. That is, the determinant of the following matrix:

\[
\text{det}\begin{pmatrix} 
4 & 0 & k \\ 
0 & 2 & 4 \\ 
2 & 4 & 2 
\end{pmatrix} = 0
\]

Now, let's calculate this determinant to find \( k \).

\[
\text{det}\begin{pmatrix} 
4 & 0 & k \\ 
0 & 2 & 4 \\ 
2 & 4 & 2 
\end{pmatrix} = 4 \begin{vmatrix} 2 & 4 \\ 4 & 2 \end{vmatrix} - 0 + k \begin{vmatrix} 0 & 2 \\ 2 & 4 \end{vmatrix}
\]

Let's calculate these minors and find \( k \). I'll compute this next.The determinant of the matrix is approximately \(-64\). This means that for \( k = 4 \), the vectors are linearly independent, as the determinant is not zero.

To find the value of \( k \) that makes the vectors linearly dependent, we need to compute the determinant with \( k \) as a variable and solve for when it equals zero. I will proceed with that calculation next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to discuss this concept in more detail, or do you have any further questions about this problem?

Here are five related questions to deepen your understanding:
1. How do you verify if a set of vectors is linearly independent or dependent?
2. What is the geometric interpretation of linearly dependent vectors?
3. How does the determinant relate to the linear independence of vectors?
4. What happens when the determinant of a matrix is non-zero?
5. How can you solve for a variable in a determinant equation?

**Tip:** A set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the others.

If p = 4, for what value of k are the vectors below linearly dependent?

\( v_1 = \begin{pmatrix} p \\ 0 \\ 2 \end{pmatrix}, v_2 = \begin{pmatrix} 0 \\ 2 \\ 4 \end{pmatrix}, v_3 = \begin{pmatrix} k \\ 4 \\ 2 \end{pmatrix} \)

Solve for k to make the vectors v_1, v_2, and v_3 linearly dependent by calculating the determinant of a 3x3 matrix. Explore concepts in linear algebra, including the use of determinants and linear dependence, with step-by-step explanation.

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