If `A = [(2,-1,1),(-2,3,-2),(-4,4,-3)]` then find `A^(2)`

Answer

To find \( A^2 \) for the matrix \( A = \begin{pmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{pmatrix} \), we will perform matrix multiplication of \( A \) with itself.

### Step-by-Step Solution:

1. **Write down the matrix multiplication**:
\[ A^2 = A \times A = \begin{pmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{pmatrix} \times \begin{pmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{pmatrix} \]

2. **Calculate the elements of the resulting matrix**:

- **Element (1,1)**:
\[ 2 \cdot 2 + (-1) \cdot (-2) + 1 \cdot (-4) = 4 + 2 - 4 = 2 \]

- **Element (1,2)**:
\[ 2 \cdot (-1) + (-1) \cdot 3 + 1 \cdot 4 = -2 - 3 + 4 = -1 \]

- **Element (1,3)**:
\[ 2 \cdot 1 + (-1) \cdot (-2) + 1 \cdot (-3) = 2 + 2 - 3 = 1 \]

- **Element (2,1)**:
\[ -2 \cdot 2 + 3 \cdot (-2) + (-2) \cdot (-4) = -4 - 6 + 8 = -2 \]

- **Element (2,2)**:
\[ -2 \cdot (-1) + 3 \cdot 3 + (-2) \cdot 4 = 2 + 9 - 8 = 3 \]

- **Element (2,3)**:
\[ -2 \cdot 1 + 3 \cdot (-2) + (-2) \cdot (-3) = -2 - 6 + 6 = -2 \]

- **Element (3,1)**:
\[ -4 \cdot 2 + 4 \cdot (-2) + (-3) \cdot (-4) = -8 - 8 + 12 = -4 \]

- **Element (3,2)**:
\[ -4 \cdot (-1) + 4 \cdot 3 + (-3) \cdot 4 = 4 + 12 - 12 = 4 \]

- **Element (3,3)**:
\[ -4 \cdot 1 + 4 \cdot (-2) + (-3) \cdot (-3) = -4 - 8 + 9 = -3 \]

3. **Combine the results into the resulting matrix**:
\[ A^2 = \begin{pmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{pmatrix} \]

### Final Result:
\[ A^2 = \begin{pmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{pmatrix} \]