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We are given the matrix equation \( A \vec{x} = \vec{b} \) where:

\[
A = \begin{bmatrix} 2 & 2 & 1 \\ 3 & -3 & -1 \\ 5 & -1 & 2 \end{bmatrix}, \quad
\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \quad
\vec{b} = \begin{bmatrix} 6 \\ 9 \\ 20 \end{bmatrix}
\]

### (i) Determine \( A^{-1} \) using cofactors

The inverse of a matrix \( A \), if it exists, is given by:
\[
A^{-1} = \frac{1}{\det(A)} \text{adj}(A)
\]
where \( \det(A) \) is the determinant of \( A \), and \( \text{adj}(A) \) is the adjugate matrix (transpose of the cofactor matrix).

#### Step 1: Compute the determinant of \( A \)

The determinant \( \det(A) \) for a 3x3 matrix is computed as:

\[
\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
\]

Let’s calculate that first.

#### Step 2: Calculate the adjugate matrix \( \text{adj}(A) \)

We will compute the cofactors and transpose the cofactor matrix to get the adjugate matrix.

### (ii) Solve \( \vec{x} \)

Once we find \( A^{-1} \), we can solve for \( \vec{x} \) by multiplying both sides of the equation by \( A^{-1} \):
\[
\vec{x} = A^{-1} \vec{b}
\]

Let me compute the determinant, adjugate matrix, and \( \vec{x} \) now.The determinant of matrix \( A \) is \( \det(A) = -24 \).

Since the determinant is non-zero, matrix \( A \) is invertible. Now, let's calculate the adjugate matrix by finding the cofactors of \( A \), and then compute \( A^{-1} \).It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the manual steps to calculate the inverse matrix and solve for \( \vec{x} \).

Would you like to continue with that, or do you have any other questions?

Here are some related questions you may find helpful:
1. How do you calculate the cofactor of a matrix element?
2. What is the adjugate matrix and how is it used to find the inverse?
3. How do you solve systems of linear equations using matrix inverses?
4. How do you know if a matrix is invertible?
5. What is Cramer's Rule and how is it used to solve linear systems?

**Tip:** When solving matrix equations, ensure the determinant of the matrix is non-zero, otherwise, the matrix doesn't have an inverse.

Given the matrix equation A * x = b, where x = [x1, x2, x3], b = [6, 9, 20], and A = [[2, 2, 1], [3, -3, -1], [5, -1, 2]], (i) determine A^(-1) using cofactors of A, (ii) solve for x.

Learn how to solve a matrix equation A * x = b using cofactors and determinants to find A^(-1). This problem involves the calculation of the determinant, adjugate matrix, and solving the system of equations. Suitable for college-level students studying linear algebra.

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