https://file.aiquickdraw.com/mathbot/file/172724268503999gzwofk.jpg

The problem asks to find the inverses of the given 3x3 matrices. We need to calculate the inverse for each matrix listed in the image:

1. **Matrix A**:
   \[
   A = \begin{bmatrix}
   3 & 0 & 1 \\
   2 & 5 & 1 \\
   -1 & 0 & 2
   \end{bmatrix}
   \]

2. **Matrix B**:
   \[
   B = \begin{bmatrix}
   1 & 4 & 2 \\
   1 & 0 & 2 \\
   2 & 1 & 4
   \end{bmatrix}
   \]

3. **Matrix C**:
   \[
   C = \begin{bmatrix}
   1 & 1 & 2 \\
   1 & 2 & 3 \\
   1 & 3 & 2
   \end{bmatrix}
   \]

4. **Matrix D**:
   \[
   D = \begin{bmatrix}
   2 & -1 & 3 \\
   -4 & 3 & -2 \\
   1 & -1 & 1
   \end{bmatrix}
   \]

5. **Matrix E**:
   \[
   E = \begin{bmatrix}
   3 & 5 & 2 \\
   5 & 7 & 4 \\
   0 & 1 & 6
   \end{bmatrix}
   \]

6. **Matrix F**:
   \[
   F = \begin{bmatrix}
   2 & -1 & 3 \\
   -4 & 3 & -2 \\
   1 & -1 & 1
   \end{bmatrix}
   \]

7. **Matrix K**:
   \[
   K = \begin{bmatrix}
   2 & 3 & 1 \\
   0 & -3 & -2 \\
   5 & -1 & 4
   \end{bmatrix}
   \]

8. **Matrix L**:
   \[
   L = \begin{bmatrix}
   2 & 5 & 4 \\
   -1 & 6 & 0 \\
   8 & 9 & 3
   \end{bmatrix}
   \]

9. **Matrix M**:
   \[
   M = \begin{bmatrix}
   3 & 0 & 11 \\
   2 & 5 & 1 \\
   -1 & 0 & 2
   \end{bmatrix}
   \]

To find the inverse of a matrix \( A \), we use the formula:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]

where \(\text{det}(A)\) is the determinant of \(A\) and \(\text{adj}(A)\) is the adjugate of \(A\).

I will start calculating the inverses one by one. Let's begin with **Matrix A**.The inverse of **Matrix A** is:

\[
A^{-1} = \begin{bmatrix}
0.2857 & 0 & -0.1429 \\
-0.1429 & 0.2 & -0.0286 \\
0.1429 & 0 & 0.4286
\end{bmatrix}
\]

Would you like the inverse for another matrix or a detailed explanation of the process? Here are some related questions for further exploration:

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?
3. What are the properties of an invertible matrix?
4. How do we interpret the inverse of a matrix in practical applications?
5. Can a matrix with a determinant of zero have an inverse?

**Tip:** To verify if a matrix is invertible, always check its determinant. If the determinant is zero, the matrix is not invertible.

Tentukan invers matriks ordo 3x3 berikut.

Pelajari cara menentukan invers dari matriks ordo 3x3 dengan contoh-contoh yang terperinci. Masalah ini meliputi konsep utama aljabar matriks, determinan, dan sifat matriks yang dapat dibalik. Cocok untuk siswa kelas 10-12.

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