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To solve this problem, we need to find the adjugate (or adjoint) of matrix \( A \) and then use it to determine the inverse \( A^{-1} \). Here is the matrix \( A \):

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

### Steps to Solve:

1. **Calculate the Determinant of \( A \)**:  
   The determinant, \( \det(A) \), is needed for both the inverse and adjugate. For a \( 3 \times 3 \) matrix, the determinant is calculated as:
   \[
   \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
   \]
   where the entries of the matrix are:
   \[
   A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}
   \]
   Substituting values from \( A \), we will find \( \det(A) \).

2. **Find the Cofactor Matrix**:  
   Calculate the cofactor of each element by taking the minor (determinant of the \( 2 \times 2 \) submatrix) and applying the sign changes.

3. **Form the Adjugate Matrix**:  
   The adjugate matrix, \( \text{adj}(A) \), is the transpose of the cofactor matrix.

4. **Calculate \( A^{-1} \)**:  
   The inverse of \( A \), if it exists, is given by:
   \[
   A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)
   \]
   This calculation is possible if \( \det(A) \neq 0 \).

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I’ll begin by calculating each step in detail.Here are the results for each step:

1. **Determinant of \( A \)**:  
   \[
   \det(A) = 17
   \]

2. **Adjugate of \( A \) (adj \( A \))**:  
   \[
   \text{adj}(A) = \begin{pmatrix} -5 & 6 & 4 \\ -2 & -1 & 5 \\ 10 & 5 & -8 \end{pmatrix}
   \]

3. **Inverse of \( A \) ( \( A^{-1} \) )**:  
   \[
   A^{-1} = \frac{1}{17} \cdot \text{adj}(A) = \begin{pmatrix} -0.294 & -0.118 & 0.588 \\ 0.353 & -0.059 & 0.294 \\ 0.235 & 0.294 & -0.471 \end{pmatrix}
   \]

Let me know if you want further details on the calculations or have any questions.

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### Related Questions:
1. How do you calculate the determinant of a \( 3 \times 3 \) matrix?
2. What are the conditions for a matrix to have an inverse?
3. How is the cofactor of a matrix element determined?
4. Why is the adjugate matrix the transpose of the cofactor matrix?
5. Can you use this method for finding the inverse of larger matrices?

### Tip:
For any square matrix \( A \), the inverse exists only if \( \det(A) \neq 0 \).

Given \( A = \begin{pmatrix} -1 & 2 & 0 \\ 4 & 0 & 5 \\ 2 & 1 & 1 \end{pmatrix} \), find \( \text{adj}(A) \) and hence find its inverse, \( A^{-1} \).

Learn how to find the adjugate and inverse of a 3x3 matrix A with a step-by-step approach. This exercise covers matrix algebra concepts including the determinant, cofactor matrix, and inverse formula. Suitable for students in Grades 11-12.

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