To find the eigenvector corresponding to the eigenvalue $\lambda = 4$ for the matrix:

$\begin{bmatrix} 1 & 0 & 0 \\ 3 & -2 & 0 \\ 2 & 3 & 4 \end{bmatrix}$

we'll proceed as follows:

Step 1: Set Up the Equation

To find the eigenvector, we need to solve: $(A - 4I)\mathbf{x} = 0$ where $A$ is the given matrix and $I$ is the identity matrix of the same dimension.

The matrix $4I$ is: $4I = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$

Now, calculate $A - 4I$: $A - 4I = \begin{bmatrix} 1 - 4 & 0 & 0 \\ 3 & -2 - 4 & 0 \\ 2 & 3 & 4 - 4 \end{bmatrix} = \begin{bmatrix} -3 & 0 & 0 \\ 3 & -6 & 0 \\ 2 & 3 & 0 \end{bmatrix}$

Step 2: Solve $(A - 4I)\mathbf{x} = 0$

We now need to solve the following system: $\begin{bmatrix} -3 & 0 & 0 \\ 3 & -6 & 0 \\ 2 & 3 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

This system gives us three equations:

1. $-3x_1 = 0 \Rightarrow x_1 = 0$
2. $3x_1 - 6x_2 = 0 \Rightarrow -6x_2 = 0 \Rightarrow x_2 = 0$
3. $2x_1 + 3x_2 = 0 \Rightarrow x_3$ can be any value (let $x_3 = -1$).

Thus, a corresponding eigenvector is: [ \mathbf{x} = \begin{bmatrix}