To compute the condition number of a matrix $A$ with respect to the 2-norm, we use the following formula:

$\kappa_2(A) = \|A\|_2 \cdot \|A^{-1}\|_2$

where $\|A\|_2$ is the 2-norm of the matrix $A$, and $\|A^{-1}\|_2$ is the 2-norm of the inverse of $A$.

Steps:

1. Find the 2-norm of matrix $A$: The 2-norm of a matrix is the largest singular value $\sigma_{\text{max}}$ of the matrix $A$.

2. Find the 2-norm of the inverse matrix $A^{-1}$: This is the largest singular value of the inverse matrix $A^{-1}$.

3. Compute the condition number: Multiply the 2-norm of $A$ by the 2-norm of $A^{-1}$.

Let's begin by computing these values for your matrix $A$:

\frac{1}{3} & \frac{2}{3} & -\frac{2}{3} \\ -\frac{2}{3} & \frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \end{bmatrix}$$ I'll compute the condition number now.The condition number of the matrix $$A$$ with respect to the 2-norm is approximately $$1$$. This indicates that the matrix $$A$$ is very well-conditioned, meaning it is stable for numerical computations and small changes in input will not lead to significant changes in the output. Would you like any further details or clarifications? Here are five related questions for further exploration: 1. How is the condition number affected by small perturbations in matrix elements? 2. What is the significance of a condition number being 1 in numerical analysis? 3. How would the condition number change if we used a different norm, such as the $$\infty$$-norm? 4. Can you find the singular values of matrix $$A$$? 5. What role does matrix conditioning play in solving linear systems? **Tip**: A condition number close to 1 means the matrix is well-conditioned, but as it increases, the matrix becomes more sensitive to numerical errors.