Question

If

\[ \begin{pmatrix} z \\ y \end{pmatrix} \]

is an eigenvector corresponding to a real eigenvalue of the matrix

\[ \begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & -4 \\ 0 & 1 & 3 \end{pmatrix} \]

then \( z - y \) is equal to .................

Hint:

When solving eigenvalue problems, express the system of equations and solve for the unknowns corresponding to the eigenvector components.

Answer 1

Correct Answer: 3

Step 1: Set up the eigenvalue equation.
We are given the matrix \( A \) and the eigenvector \( \begin{pmatrix} z \\ y \end{pmatrix} \). The eigenvalue equation is:

\[ A \begin{pmatrix} z \\ y \end{pmatrix} = \lambda \begin{pmatrix} z \\ y \end{pmatrix}. \]

Step 2: Solve for \( z \) and \( y \).
Substituting the matrix \( A \) and the eigenvector, we get the system of equations:

\[ \begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & -4 \\ 0 & 1 & 3 \end{pmatrix} \begin{pmatrix} z \\ y \end{pmatrix} = \lambda \begin{pmatrix} z \\ y \end{pmatrix}. \]

By solving this system, we find that \( z - y = 0 \).
Step 3: Conclusion.
Therefore, \( z - y \) is:

\[ \boxed{0}. \]