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The given matrix \( A \) is:

\[
A = \begin{bmatrix} 
0 & 1 & \frac{2}{3} \\ 
1 & 2 & \frac{2}{3} \\ 
3 & 1 & 1 
\end{bmatrix}
\]

To find the inverse of matrix \( A \), we can use one of the following methods:

1. **Adjugate (Cofactor) Method**: This involves finding the determinant of the matrix, the matrix of minors, cofactors, adjugate, and finally dividing by the determinant.
2. **Gaussian Elimination**: This involves augmenting the matrix with the identity matrix and performing row operations to reduce it to the identity matrix, turning the augmented identity matrix into the inverse.

I will start by calculating the determinant of matrix \( A \), and if the determinant is non-zero, proceed to find the inverse.

Let's begin.The determinant of matrix \( A \) is approximately \( -2.33 \), which is non-zero. Therefore, the matrix \( A \) is invertible, and we can proceed to calculate its inverse.

I'll compute the inverse of the matrix next.The inverse of the matrix \( A \) is:

\[
A^{-1} = \begin{bmatrix}
-0.571 & 0.143 & 0.286 \\
-0.429 & 0.857 & -0.286 \\
2.143 & -1.286 & 0.429
\end{bmatrix}
\]

Would you like further details on how the inverse was calculated or any other aspect of this solution?

Here are some related questions to expand your understanding:
1. What is the role of the determinant in determining the invertibility of a matrix?
2. Can a matrix with a determinant of zero have an inverse?
3. How would you perform matrix inversion manually using Gaussian elimination?
4. What is the significance of matrix inversion in solving systems of equations?
5. How can matrix operations be applied to real-world problems?

**Tip**: When calculating the inverse of a matrix manually, always verify that the determinant is non-zero; otherwise, the matrix does not have an inverse.

Find the inverse of the matrix A = [[0, 1, 2/3], [1, 2, 2/3], [3, 1, 1]]

Learn how to calculate the inverse of the 3x3 matrix A = [[0, 1, 2/3], [1, 2, 2/3], [3, 1, 1]] using determinant and cofactor methods. This problem covers key concepts in linear algebra, including matrix inversion and determinants. Suitable for advanced high school and undergraduate students.

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