1. **Problem:** Find the inverse of the matrix $A$ using the adjoint method. 2. **Formula:** The inverse of a matrix $A$ is given by $$A^{-1} = \frac{1}{|A|} \times \text{adj}(A)$$ where $|A|$ is the determinant of $A$ and $\text{adj}(A)$ is the adjoint (transpose of cofactor matrix). 3. **Step 1: Calculate the Determinant $|A|$** Calculate determinant to ensure it is non-zero: $$|A| = 1(4 - 2) - 0 + 1(-1 - 2) = 2 - 3 = -1$$ Since $|A| = -1 \neq 0$, inverse exists. 4. **Step 2: Matrix of Cofactors $C$** Calculate each cofactor: $$C = \begin{bmatrix} 2 & 4 & -3 \\ 1 & 1 & -1 \\ -2 & -3 & 2 \end{bmatrix}$$ 5. **Step 3: Adjoint Matrix $\text{adj}(A)$** Transpose the cofactor matrix: $$\text{adj}(A) = C^T = \begin{bmatrix} 2 & 1 & -2 \\ 4 & 1 & -3 \\ -3 & -1 & 2 \end{bmatrix}$$ 6. **Step 4: Calculate the Inverse** Multiply adjoint by $1/|A| = -1$: $$A^{-1} = -1 \times \begin{bmatrix} 2 & 1 & -2 \\ 4 & 1 & -3 \\ -3 & -1 & 2 \end{bmatrix} = \begin{bmatrix} -2 & -1 & 2 \\ -4 & -1 & 3 \\ 3 & 1 & -2 \end{bmatrix}$$ **Final Answer:** $$A^{-1} = \begin{bmatrix} -2 & -1 & 2 \\ -4 & -1 & 3 \\ 3 & 1 & -2 \end{bmatrix}$$