The matrix A = `[{:(2 , -2, -4),(- 1, 3, 4),(1 , - 2, -3):}]` is

Options

A. Nilpotent
B. Idempotent
C. Orthogonal
D. Involutary

Answer

To determine the nature of the matrix \( A = \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix} \), we will check if it is nilpotent, idempotent, orthogonal, or involuntary. We will do this by calculating \( A^2 \) and comparing it with \( A \).

### Step-by-Step Solution:

1. **Calculate \( A^2 \)**:
To find \( A^2 \), we multiply matrix \( A \) by itself:
\[ A^2 = A \times A = \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix} \times \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix} \]

2. **Matrix Multiplication**:
We will multiply the rows of the first matrix by the columns of the second matrix.

- **First Row**:
- First element: \( 2 \times 2 + (-2) \times (-1) + (-4) \times 1 = 4 + 2 - 4 = 2 \)
- Second element: \( 2 \times (-2) + (-2) \times 3 + (-4) \times (-2) = -4 - 6 + 8 = -2 \)
- Third element: \( 2 \times (-4) + (-2) \times 4 + (-4) \times (-3) = -8 - 8 + 12 = -4 \)

- **Second Row**:
- First element: \( -1 \times 2 + 3 \times (-1) + 4 \times 1 = -2 - 3 + 4 = -1 \)
- Second element: \( -1 \times (-2) + 3 \times 3 + 4 \times (-2) = 2 + 9 - 8 = 3 \)
- Third element: \( -1 \times (-4) + 3 \times 4 + 4 \times (-3) = 4 + 12 - 12 = 4 \)

- **Third Row**:
- First element: \( 1 \times 2 + (-2) \times (-1) + (-3) \times 1 = 2 + 2 - 3 = 1 \)
- Second element: \( 1 \times (-2) + (-2) \times 3 + (-3) \times (-2) = -2 - 6 + 6 = -2 \)
- Third element: \( 1 \times (-4) + (-2) \times 4 + (-3) \times (-3) = -4 - 8 + 9 = -3 \)

Thus, we have:
\[ A^2 = \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix} \]

3. **Compare \( A^2 \) with \( A \)**:
We can see that:
\[ A^2 = A \]
This means that the matrix \( A \) is idempotent.

### Conclusion:
The matrix \( A \) is idempotent.