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The image appears to include a matrix operation and a question involving the calculation of the determinant of a 3x3 matrix. Below is a detailed solution to the problem:

### Matrix Determinant Calculation
Given the matrix \( A \):

\[ A = \begin{pmatrix}
1 & 4 & 3 \\
-1 & 5 & 2 \\
2 & 1 & 1 \\
\end{pmatrix} \]

We need to find the determinant of this matrix, denoted as \( \det(A) \).

### Step-by-Step Solution:

1. **Expand along the first row:**

\[ \det(A) = 1 \cdot \det \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix} - 4 \cdot \det \begin{pmatrix} -1 & 2 \\ 2 & 1 \end{pmatrix} + 3 \cdot \det \begin{pmatrix} -1 & 5 \\ 2 & 1 \end{pmatrix} \]

2. **Calculate the determinants of the 2x2 matrices:**

   - For \(\det \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix}\):

     \[
     \det \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix} = (5 \cdot 1) - (2 \cdot 1) = 5 - 2 = 3
     \]

   - For \(\det \begin{pmatrix} -1 & 2 \\ 2 & 1 \end{pmatrix}\):

     \[
     \det \begin{pmatrix} -1 & 2 \\ 2 & 1 \end{pmatrix} = (-1 \cdot 1) - (2 \cdot 2) = -1 - 4 = -5
     \]

   - For \(\det \begin{pmatrix} -1 & 5 \\ 2 & 1 \end{pmatrix}\):

     \[
     \det \begin{pmatrix} -1 & 5 \\ 2 & 1 \end{pmatrix} = (-1 \cdot 1) - (5 \cdot 2) = -1 - 10 = -11
     \]

3. **Substitute these values back into the determinant formula:**

\[ \det(A) = 1 \cdot 3 - 4 \cdot (-5) + 3 \cdot (-11) \]

4. **Simplify the expression:**

\[ \det(A) = 3 + 20 - 33 = -10 \]

So, the determinant of the matrix \( A \) is \(\boxed{-10}\).

Would you like more details on any part of this solution, or do you have any questions?

### Related Questions:

1. What is the geometric interpretation of the determinant of a matrix?
2. How does the determinant of a matrix change when two rows are swapped?
3. What is the determinant of a triangular matrix?
4. How is the determinant affected by scaling one row of a matrix?
5. Can the determinant be used to determine if a matrix is invertible?
6. How would you find the determinant of a 4x4 matrix?
7. What is the cofactor expansion method for calculating determinants?
8. How does row reduction affect the calculation of the determinant?

### Tip:
When calculating determinants, it's often helpful to choose a row or column with the most zeros to simplify the computation.

Learn how to calculate the determinant of a 3x3 matrix with a detailed step-by-step solution. This problem covers matrix operations and the determinant formula, suitable for advanced high school students and beyond.

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