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The question asks for the inverse of the matrix \( A \), where:

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

We are tasked with finding the inverse of matrix \( A \), and then matching it with one of the options provided (A, B, C, D, or E).

### Step 1: Find the Determinant of \( A \)

The determinant of a 3x3 matrix \( A \) is calculated as:

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

where the elements of matrix \( A \) are as follows:

\[
A = \begin{pmatrix} 
a & b & c \\ 
d & e & f \\ 
g & h & i 
\end{pmatrix}
= \begin{pmatrix} 
2 & -1 & 1 \\ 
-1 & 1 & 1 \\ 
3 & -1 & 2 
\end{pmatrix}
\]

Thus:
- \( a = 2 \)
- \( b = -1 \)
- \( c = 1 \)
- \( d = -1 \)
- \( e = 1 \)
- \( f = 1 \)
- \( g = 3 \)
- \( h = -1 \)
- \( i = 2 \)

Now we calculate the determinant. I'll handle the determinant and inversion calculation now.The determinant of matrix \( A \) is \( -1 \), which means the matrix is invertible. The inverse of matrix \( A \) is:

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

This matches option **A**.

Would you like any further details or explanations?

Here are 5 related questions you might find interesting:
1. How is the determinant of a 3x3 matrix calculated?
2. What properties do invertible matrices have?
3. Can a matrix with a zero determinant have an inverse?
4. How is matrix multiplication used to verify if a matrix is the inverse of another?
5. What real-world applications involve finding matrix inverses?

**Tip:** When working with matrix inverses, always check the determinant first! If it's zero, the matrix doesn't have an inverse.

Find the inverse of the matrix A = [[2, -1, 1], [-1, 1, 1], [3, -1, 2]]. Choose the correct inverse from the options A, B, C, D, or E.

Learn how to find the inverse of the 3x3 matrix A = [[2, -1, 1], [-1, 1, 1], [3, -1, 2]] using the determinant and adjugate method. Step-by-step solution for high school or undergraduate students, including the key formulas and theorems for matrix inversion.

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