Question

\[ \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \] is an eigenvector of the matrix \[ \begin{pmatrix} 3 & 6 & 7 \\ 3 & 3 & 7 \\ 5 & 6 & 5 \end{pmatrix}. \] With respect to which eigenvalue?

• A. 15
• B. 2
• C. -15
• D. -2

Hint:

To quickly find the eigenvalue when given an eigenvector, simply compute the first row of the matrix-vector multiplication $Mv$ and divide it by the first component of the vector $v$. Here: $-2 / 1 = -2$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
By definition, if $v$ is an eigenvector of a square matrix $M$ corresponding to an eigenvalue $\lambda$, they satisfy the equation:
\[ M v = \lambda v \]

Step 2: Detailed Explanation:
Let us denote:
\[ M = \begin{pmatrix} 3 & 6 & 7 3 & 3 & 7 5 & 6 & 5 \end{pmatrix} \quad \text{and} \quad v = \begin{pmatrix} 1 -2 1 \end{pmatrix} \]
Compute the product $M v$:
\[ M v = \begin{pmatrix} 3 & 6 & 7 3 & 3 & 7 5 & 6 & 5 \end{pmatrix} \begin{pmatrix} 1 -2 1 \end{pmatrix} = \begin{pmatrix} 3 \cdot (1) + 6 \cdot (-2) + 7 \cdot (1) 3 \cdot (1) + 3 \cdot (-2) + 7 \cdot (1) 5 \cdot (1) + 6 \cdot (-2) + 5 \cdot (1) \end{pmatrix} \]
Calculate the individual row sums:
- Row 1: $3 - 12 + 7 = -2$
- Row 2: $3 - 6 + 7 = 4$
- Row 3: $5 - 12 + 5 = -2$
So we have:
\[ M v = \begin{pmatrix} -2 4 -2 \end{pmatrix} \]
Now express $M v$ as a scalar multiple of the original vector $v$:
\[ M v = \begin{pmatrix} -2 4 -2 \end{pmatrix} = -2 \begin{pmatrix} 1 -2 1 \end{pmatrix} \]
Comparing this with the definition $M v = \lambda v$, we find:
\[ \lambda = -2 \]
Therefore, the corresponding eigenvalue is $-2$.

Step 3: Final Answer
The correct option is (D).