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: Eigenvalue calculation.
We are given the matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}\) and the eigenvector \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\). The general equation for an eigenvalue is: \[ A \mathbf{v} = \lambda \mathbf{v} \] Where \(A\) is the matrix, \(\mathbf{v}\) is the eigenvector, and \(\lambda\) is the eigenvalue.

Step 2: Apply the matrix multiplication.
Multiplying the matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}\) by the eigenvector \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\), we get: \[ \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 4 \times 1 + 2 \times 2 \\ 2 \times 1 + 4 \times 2 \end{bmatrix} = \begin{bmatrix} 8 \\ 10 \end{bmatrix} \]

Step 3: Compare the result. \\ The result \(\begin{bmatrix} 8 \\ 10 \end{bmatrix}\) is a scalar multiple of the original eigenvector \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\). The scalar multiple is \(4\), which means the eigenvalue is \(4\).

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