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We are tasked with finding the value of \( k \) such that the vector \( \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) is an eigenvector of the matrix:

\[
A = \begin{pmatrix} 4 & 1 & 2 \\ k & 2 & 1 \\ 14 & -4 & 10 \end{pmatrix}
\]

Given that the vector \( \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) is an eigenvector of the matrix, the matrix equation holds:

\[
A \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \lambda \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}
\]

Where \( \lambda \) is the eigenvalue corresponding to the eigenvector. Let's compute this to find \( k \). I will first calculate the matrix-vector product on the left-hand side.For \( k = 17 \), the matrix-vector product is:

\[
A \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 24 \\ 36 \end{pmatrix}
\]

This is exactly \( 12 \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \), which means that the eigenvalue \( \lambda = 12 \).

Thus, the value of \( k \) that makes \( \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) an eigenvector is indeed \( k = 17 \).

Would you like more details on the steps? Let me know if you have any questions!

Here are some related questions to expand your understanding:
1. How can we find eigenvalues and eigenvectors for a general matrix?
2. What is the significance of the eigenvalue in the context of transformations?
3. How would the answer change if the eigenvector had been different?
4. Can a matrix have more than one eigenvector corresponding to the same eigenvalue?
5. What properties do symmetric matrices have concerning their eigenvalues and eigenvectors?

**Tip:** Eigenvalues and eigenvectors have important applications in fields like machine learning, quantum mechanics, and vibration analysis.

The value of k such that the vector [1, 2, 3] is an eigenvector of the matrix [4, 1, 2; k, 2, 1; 14, -4, 10].

Solve the problem to find the value of k such that the vector [1, 2, 3] is an eigenvector of the matrix [4, 1, 2; k, 2, 1; 14, -4, 10]. This problem covers key concepts of linear algebra, including eigenvalues and eigenvectors. Suitable for undergraduate students.

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