Finding the Eigenvalue for a Given Matrix and Eigenvector

The question asks for the eigenvalue corresponding to a specific eigenvector for a given matrix. We are provided with the matrix $A = \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}$ and the eigenvector $v = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$.

Understanding Eigenvalues and Eigenvectors

An eigenvalue ($\lambda$) and its corresponding non-zero eigenvector ($v$) of a square matrix ($A$) satisfy the equation:

$Av = \lambda v$

This equation means that when the matrix $A$ acts on the eigenvector $v$, the result is simply the eigenvector scaled by a factor $\lambda$. This scaling factor $\lambda$ is the eigenvalue.

Step-by-Step Calculation

To find the eigenvalue $\lambda$, we need to compute the product of the matrix $A$ and the eigenvector $v$, and then equate it to $\lambda v$.

Step 1: Calculate the Matrix-Vector Product ($Av$)

We multiply the given matrix $A$ by the given eigenvector $v$:

$Av = \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix}$

Performing the matrix multiplication:

$ Av = \begin{bmatrix} (4 \times 1) + (2 \times 1) \\ (2 \times 1) + (4 \times 1) \end{bmatrix} $ $ Av = \begin{bmatrix} 4 + 2 \\ 2 + 4 \end{bmatrix} $ $ Av = \begin{bmatrix} 6 \\ 6 \end{bmatrix} $

Step 2: Relate the Result to $\lambda v$

Now, we use the eigenvalue equation $Av = \lambda v$:

$ \begin{bmatrix} 6 \\ 6 \end{bmatrix} = \lambda \begin{bmatrix} 1 \\ 1 \end{bmatrix} $

Distributing the scalar $\lambda$ on the right side:

$ \begin{bmatrix} 6 \\ 6 \end{bmatrix} = \begin{bmatrix} \lambda \times 1 \\ \lambda \times 1 \end{bmatrix} $ $ \begin{bmatrix} 6 \\ 6 \end{bmatrix} = \begin{bmatrix} \lambda \\ \lambda \end{bmatrix} $

Step 3: Determine the Eigenvalue ($\lambda$)

By comparing the corresponding elements of the resulting vectors, we can determine the value of $\lambda$:

• From the first element: $6 = \lambda$
• From the second element: $6 = \lambda$

Both comparisons yield the same value for $\lambda$.

Therefore, the eigenvalue corresponding to the eigenvector $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ for the matrix $\begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}$ is 6.