The power method is an iterative technique used to approximate the dominant (largest) eigenvalue of a matrix. Let's outline the steps and compute the dominant eigenvalue for the given matrix:

Matrix:

$A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

Step-by-Step Solution:

1. Choose an initial vector $x^{(0)}$.

   • Typically, we choose $x^{(0)} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$.

2. Normalize the initial vector $x^{(0)}$.

   • Compute the norm of $x^{(0)}$:
     $\| x^{(0)} \| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$
   • Normalize:
     $x^{(0)} = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$

3. Iteratively compute the matrix-vector product $y^{(k)} = A x^{(k)}$ until convergence.

   • At each step, update the vector:
     $x^{(k+1)} = \frac{y^{(k)}}{\| y^{(k)} \|}$
   • Approximate the dominant eigenvalue as:
     $\lambda^{(k)} = \frac{y^{(k)} \cdot x^{(k)}}{x^{(k)} \cdot x^{(k)}}$

4. Continue iterating until $x^{(k)}$ converges to a fixed vector, or until $\lambda^{(k)}$ stabilizes.

Let's solve this problem using these steps.

I'll implement the power method to find the dominant eigenvalue for matrix $A$.