Question

If \[ \begin{bmatrix} 1 1 \end{bmatrix} \] is an eigenvector of the matrix \[ A= \begin{bmatrix} m& 3 \\ 3& m \end{bmatrix}, \] then the corresponding eigenvalue is:

• A. $m-3$
• B. $m+3$
• C. $m$
• D. $3m$

Hint:

If all components scale equally after multiplication, that scale factor is the eigenvalue.

Answer 1

The Correct Option is B

Concept: If $\vec{x}$ is an eigenvector of matrix $A$, then: \[ A\vec{x}=\lambda \vec{x} \] This means multiplication of matrix with vector gives a scalar multiple of the same vector.

Step 1: Write matrix-vector multiplication.
\[ A\vec{x}= \begin{bmatrix} m& 3 \\ 3& m \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \] Now multiply: First row: \[ m(1)+3(1)=m+3 \] Second row: \[ 3(1)+m(1)=m+3 \] So, \[ A\vec{x}= \begin{bmatrix} m+3 m+3 \end{bmatrix} \]

Step 2: Compare with eigenvector form.
\[ \begin{bmatrix} m+3 \\ m+3 \end{bmatrix} = (m+3) \begin{bmatrix} 1 1 \end{bmatrix} \] Thus eigenvalue: \[ \lambda = m+3 \]