The inverse matrix of `A=[(0,1,2),(1,2,3),(3,1,1)]` is

Options

A. `[(1/2,-1/2,1/2),(-4,3,-1),(5/2,-3/2,1/2)]`
B. `[(1/2,-4,5/4),(1,-6,3),(1,2,-1)]`
C. `1/2[(1,2,3),(3,2,1),(4,2,3)]`
D. `1/2[(1,-1,-1),(-8,6,-2),(5,-3,1)]`

Explanation

To find the inverse of the matrix \( A = \begin{pmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A

The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) can be calculated using the formula:

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

For our matrix \( A \):

- \( a = 0, b = 1, c = 2 \)
- \( d = 1, e = 2, f = 3 \)
- \( g = 3, h = 1, i = 1 \)

Substituting these values into the determinant formula:

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

Calculating each term:

1. \( 0(2 - 3) = 0 \)
2. \( -1(1 - 9) = -1 \cdot (-8) = 8 \)
3. \( 2(1 - 6) = 2 \cdot (-5) = -10 \)

Now, combine these results:

\[ \text{det}(A) = 0 + 8 - 10 = -2 \]

### Step 2: Calculate the Adjoint of Matrix A

The adjoint of a matrix is the transpose of the cofactor matrix. We will calculate the cofactors for each element of \( A \).

1. **Cofactor of \( a_{11} = 0 \)**:
\[ C_{11} = \text{det} \begin{pmatrix} 2 & 3 \\ 1 & 1 \end{pmatrix} = (2 \cdot 1 - 3 \cdot 1) = -1 \]

2. **Cofactor of \( a_{12} = 1 \)**:
\[ C_{12} = -\text{det} \begin{pmatrix} 1 & 3 \\ 3 & 1 \end{pmatrix} = - (1 \cdot 1 - 3 \cdot 3) = -(-8) = 8 \]

3. **Cofactor of \( a_{13} = 2 \)**:
\[ C_{13} = \text{det} \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} = (1 \cdot 1 - 2 \cdot 3) = -5 \]

4. **Cofactor of \( a_{21} = 1 \)**:
\[ C_{21} = -\text{det} \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} = -(-5) = 5 \]

5. **Cofactor of \( a_{22} = 2 \)**:
\[ C_{22} = \text{det} \begin{pmatrix} 0 & 2 \\ 3 & 1 \end{pmatrix} = (0 \cdot 1 - 2 \cdot 3) = -6 \]

6. **Cofactor of \( a_{23} = 3 \)**:
\[ C_{23} = -\text{det} \begin{pmatrix} 0 & 1 \\ 3 & 1 \end{pmatrix} = -(-3) = 3 \]

7. **Cofactor of \( a_{31} = 3 \)**:
\[ C_{31} = \text{det} \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} = (1 \cdot 3 - 2 \cdot 2) = -1 \]

8. **Cofactor of \( a_{32} = 1 \)**:
\[ C_{32} = -\text{det} \begin{pmatrix} 0 & 2 \\ 1 & 3 \end{pmatrix} = -(0 \cdot 3 - 2 \cdot 1) = 2 \]

9. **Cofactor of \( a_{33} = 1 \)**:
\[ C_{33} = \text{det} \begin{pmatrix} 0 & 1 \\ 1 & 2 \end{pmatrix} = (0 \cdot 2 - 1 \cdot 1) = -1 \]

Now, we can form the cofactor matrix:

\[ C = \begin{pmatrix} -1 & 8 & -5 \\ 5 & -6 & 3 \\ -1 & 2 & -1 \end{pmatrix} \]

Now, we take the transpose of the cofactor matrix to get the adjoint:

\[ \text{adj}(A) = C^T = \begin{pmatrix} -1 & 5 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix} \]

### Step 3: Calculate the Inverse of Matrix A

The inverse of matrix \( A \) is given by:

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

Substituting the values we have:

\[ A^{-1} = \frac{1}{-2} \cdot \begin{pmatrix} -1 & 5 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix} \]

Calculating the inverse:

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

### Final Answer

The inverse matrix of \( A \) is:

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

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