https://file.aiquickdraw.com/mathbot/file/1726180869195rbck3qwn.jpg

The problem in the image asks to determine the inverse of the given matrix \( A \), where:

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

### Steps to find the inverse of a 3x3 matrix:

Given a matrix \( A \), the inverse \( A^{-1} \) can be calculated using the formula:

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

Where:
- \( \text{det}(A) \) is the determinant of matrix \( A \),
- \( \text{adj}(A) \) is the adjugate (transpose of the cofactor matrix).

#### Step 1: Calculate the determinant of matrix \( A \)

The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is given by:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For matrix \( A = \begin{pmatrix} -1 & 2 & 1 \\ 2 & 0 & 1 \\ 3 & 0 & -1 \end{pmatrix} \):

\[
\text{det}(A) = (-1)((0)(-1) - (1)(0)) - 2((2)(-1) - (1)(3)) + 1((2)(0) - (0)(3))
\]

Simplifying:

\[
\text{det}(A) = (-1)(0 - 0) - 2(-2 - 3) + 1(0 - 0)
\]
\[
\text{det}(A) = 0 - 2(-5) + 0 = 10
\]

Thus, \( \text{det}(A) = 10 \).

#### Step 2: Find the cofactor matrix

We calculate the cofactor matrix by finding the determinant of each 2x2 minor of \( A \), changing signs according to the cofactor matrix sign convention.

For matrix \( A \):

1. The cofactor of element \( A_{11} \) (at position (1,1)) is:
   \[
   \text{Cofactor}(A_{11}) = \text{det} \begin{pmatrix} 0 & 1 \\ 0 & -1 \end{pmatrix} = (0)(-1) - (1)(0) = 0
   \]

2. The cofactor of element \( A_{12} \) (at position (1,2)) is:
   \[
   \text{Cofactor}(A_{12}) = -\text{det} \begin{pmatrix} 2 & 1 \\ 3 & -1 \end{pmatrix} = -((2)(-1) - (1)(3)) = -(-2 - 3) = 5
   \]

3. The cofactor of element \( A_{13} \) (at position (1,3)) is:
   \[
   \text{Cofactor}(A_{13}) = \text{det} \begin{pmatrix} 2 & 0 \\ 3 & 0 \end{pmatrix} = (2)(0) - (0)(3) = 0
   \]

Continuing similarly for all elements, the cofactor matrix is:

\[
\text{Cofactor}(A) = \begin{pmatrix} 0 & 5 & 0 \\ 2 & -5 & -2 \\ 2 & 5 & -4 \end{pmatrix}
\]

#### Step 3: Transpose the cofactor matrix (Adjugate matrix)

Taking the transpose of the cofactor matrix:

\[
\text{adj}(A) = \begin{pmatrix} 0 & 2 & 2 \\ 5 & -5 & 5 \\ 0 & -2 & -4 \end{pmatrix}
\]

#### Step 4: Calculate the inverse matrix

Now, we use the formula:

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

Since \( \text{det}(A) = 10 \), we have:

\[
A^{-1} = \frac{1}{10} \begin{pmatrix} 0 & 2 & 2 \\ 5 & -5 & 5 \\ 0 & -2 & -4 \end{pmatrix}
\]

Simplifying:

\[
A^{-1} = \begin{pmatrix} 0 & \frac{1}{5} & \frac{1}{5} \\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ 0 & -\frac{1}{5} & -\frac{2}{5} \end{pmatrix}
\]

### Final Answer:

\[
A^{-1} = \begin{pmatrix} 0 & \frac{1}{5} & \frac{1}{5} \\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\ 0 & -\frac{1}{5} & -\frac{2}{5} \end{pmatrix}
\]

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### Related questions:
1. What are the properties of an invertible matrix?
2. How do you find the inverse of a matrix using row operations?
3. What is the relationship between the determinant and invertibility?
4. How does matrix inversion relate to systems of linear equations?
5. Can the inverse of a singular matrix be found? Why or why not?

**Tip:** Always check if the determinant of a matrix is non-zero before trying to find its inverse; if it’s zero, the matrix has no inverse.

Tentukan invers matriks A = \begin{pmatrix} -1 & 2 & 1 \\ 2 & 0 & 1 \\ 3 & 0 & -1 \end{pmatrix}

Learn how to find the inverse of the matrix A = \begin{pmatrix} -1 & 2 & 1 \\ 2 & 0 & 1 \\ 3 & 0 & -1 \end{pmatrix} using the determinant and adjugate method. This problem covers key matrix algebra concepts, such as calculating the determinant and constructing the cofactor matrix, suitable for advanced high school or undergraduate students.

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