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The problem provided involves finding the adjoint and inverse of two matrices. Let's go step by step to address both parts of the problem.

### 1. **Find the Adjoint of the Matrices**

#### a. Matrix \( A \)
\[
A = \begin{pmatrix} 
\frac{2}{3} & -\frac{1}{3} \\
-3 & 7 
\end{pmatrix}
\]

The adjoint (or adjugate) of a 2x2 matrix is found by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. For the matrix \( A \), the adjoint is:

\[
\text{Adj}(A) = \begin{pmatrix} 
7 & \frac{1}{3} \\
3 & \frac{2}{3} 
\end{pmatrix}
\]

#### b. Matrix \( B \)
\[
B = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix}
\]

The adjoint of a 3x3 matrix requires calculating the cofactor matrix and then transposing it. However, matrix \( B \) is a singular matrix (determinant is zero), so its adjoint would involve computing cofactors that will all result in zero, meaning:

\[
\text{Adj}(B) = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]

### 2. **Find the Inverse of the Matrices**

The inverse of a matrix \( A \) is given by:
\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A)
\]

#### a. Matrix \( A \)
\[
A = \begin{pmatrix} 
1 & 2 \\
3 & 4 
\end{pmatrix}
\]
First, compute the determinant:
\[
\text{det}(A) = (1 \times 4) - (2 \times 3) = 4 - 6 = -2
\]

Then, the adjoint is found by swapping the diagonal elements and negating the off-diagonal elements:
\[
\text{Adj}(A) = \begin{pmatrix} 
4 & -2 \\
-3 & 1 
\end{pmatrix}
\]

Thus, the inverse is:
\[
A^{-1} = \frac{1}{-2} \cdot \begin{pmatrix} 
4 & -2 \\
-3 & 1 
\end{pmatrix} = \begin{pmatrix} 
-2 & 1 \\
\frac{3}{2} & -\frac{1}{2} 
\end{pmatrix}
\]

#### b. Matrix \( B \)
\[
B = \begin{pmatrix} 
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 
\end{pmatrix}
\]
As mentioned earlier, matrix \( B \) is singular since its determinant is zero. Therefore, the inverse does not exist for matrix \( B \).

---

Would you like further details or explanations? Let me know if you'd like to expand on any of these topics.

Here are 5 related questions:
1. How is the adjoint of a matrix related to its inverse?
2. What is the cofactor expansion method for larger matrices?
3. How can we identify if a matrix is singular without calculating the determinant?
4. What are the properties of a matrix that has no inverse?
5. How is the inverse of a matrix used in solving systems of equations?

**Tip**: A matrix is invertible only if its determinant is non-zero. For 2x2 matrices, this is easy to compute by using a simple formula.

Tentukan Adjoin dari matriks dibawah ini dan Tentukan Inverse dari matriks berikut.

Learn how to find the adjoint and inverse of 2x2 and 3x3 matrices with step-by-step solutions. Understand the process for calculating the determinant, adjoint, and inverse of matrices. Suitable for advanced high school and college-level students.

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