Question

The eigen value corresponding to the eigen vector \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\) for the matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}\) is:

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

When calculating the eigenvalue, the matrix multiplied by the eigenvector should result in a scalar multiple of the original eigenvector.

Answer 1

The Correct Option is B

Step 1: Calculate the eigenvalue.
Given matrix \(A = \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}\) and eigenvector \(\mathbf{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\). The eigenvalue equation is \(A \mathbf{v} = \lambda \mathbf{v}\).

Step 2: Perform matrix-vector multiplication.
Multiply the matrix by the eigenvector: \[ \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 4(1) + 2(2) \\ 2(1) + 4(2) \end{bmatrix} = \begin{bmatrix} 8 \\ 10 \end{bmatrix} \]

Step 3: Determine the eigenvalue.
The result \(\begin{bmatrix} 8 \\ 10 \end{bmatrix}\) is a scalar multiple of \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\). By comparing \(\begin{bmatrix} 8 \\ 10 \end{bmatrix}\) with \(\lambda \begin{bmatrix} 1 \\ 2 \end{bmatrix}\), we find \(\lambda = 4\).

Final Answer: \[ \boxed{4} \]