https://file.aiquickdraw.com/mathbot/file/1726622262335doiwiixi.jpg

The matrix \( A \) given in the image is:

\[
A = \begin{pmatrix} 
4 & 2 & 2 \\ 
2 & 0 & 1 \\ 
2 & 3 & 2 
\end{pmatrix}
\]

You are asked to find the adjugates (also known as cofactors) of elements \( A_{23} \) and \( A_{32} \).

1. **Cofactor of \( A_{23} \) (element in row 2, column 3)**:
   - Remove the second row and third column to get the minor matrix:
   
     \[
     \begin{pmatrix} 
     4 & 2 \\ 
     2 & 3 
     \end{pmatrix}
     \]
     
   - The determinant of this 2x2 matrix is:
   
     \[
     (4 \times 3) - (2 \times 2) = 12 - 4 = 8
     \]
     
   - The cofactor of \( A_{23} \) is \( (-1)^{2+3} \times 8 = -8 \).

2. **Cofactor of \( A_{32} \) (element in row 3, column 2)**:
   - Remove the third row and second column to get the minor matrix:
   
     \[
     \begin{pmatrix} 
     4 & 2 \\ 
     2 & 1 
     \end{pmatrix}
     \]
     
   - The determinant of this 2x2 matrix is:
   
     \[
     (4 \times 1) - (2 \times 2) = 4 - 4 = 0
     \]
     
   - The cofactor of \( A_{32} \) is \( (-1)^{3+2} \times 0 = 0 \).

### Summary:
- The cofactor of \( A_{23} \) is \( -8 \).
- The cofactor of \( A_{32} \) is \( 0 \).

Do you want more details or have any questions? Here are 5 related questions you might want to explore:
1. How do you calculate the determinant of a 3x3 matrix?
2. What is the adjugate matrix, and how is it related to the inverse of a matrix?
3. How are cofactors used in solving systems of linear equations?
4. Can a matrix with a zero determinant be invertible?
5. What are the steps to find the inverse of a matrix using the adjugate matrix?

**Tip**: Cofactors are an essential part of finding the determinant and inverse of larger matrices!

Diketahui matriks A = [4 2 2; 2 0 1; 2 3 2]. Tentukan ajudan dari A23 dan A32.

Learn how to calculate the cofactors A23 and A32 from matrix A [4 2 2; 2 0 1; 2 3 2]. This step-by-step solution covers matrix algebra concepts, including how to find the determinant of a 2x2 matrix and cofactor expansion. Suitable for undergraduate students in linear algebra.

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