To determine the properties of the matrix \( A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} \), we will check if it is an idempotent matrix, nilpotent matrix, involutary matrix, or orthogonal matrix.
### Step 1: Check if \( A \) is an idempotent matrix
An idempotent matrix satisfies the condition \( A^2 = A \).
**Calculation of \( A^2 \):**
\[
A^2 = A \cdot A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} \cdot \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}
\]
Calculating the elements of \( A^2 \):
- First row, first column: \( 2 \cdot 2 + (-2) \cdot (-1) + (-4) \cdot 1 = 4 + 2 - 4 = 2 \)
- First row, second column: \( 2 \cdot (-2) + (-2) \cdot 3 + (-4) \cdot (-2) = -4 - 6 + 8 = -2 \)
- First row, third column: \( 2 \cdot (-4) + (-2) \cdot 4 + (-4) \cdot (-3) = -8 - 8 + 12 = -4 \)
- Second row, first column: \( -1 \cdot 2 + 3 \cdot (-1) + 4 \cdot 1 = -2 - 3 + 4 = -1 \)
- Second row, second column: \( -1 \cdot (-2) + 3 \cdot 3 + 4 \cdot (-2) = 2 + 9 - 8 = 3 \)
- Second row, third column: \( -1 \cdot (-4) + 3 \cdot 4 + 4 \cdot (-3) = 4 + 12 - 12 = 4 \)
- Third row, first column: \( 1 \cdot 2 + (-2) \cdot (-1) + (-3) \cdot 1 = 2 + 2 - 3 = 1 \)
- Third row, second column: \( 1 \cdot (-2) + (-2) \cdot 3 + (-3) \cdot (-2) = -2 - 6 + 6 = -2 \)
- Third row, third column: \( 1 \cdot (-4) + (-2) \cdot 4 + (-3) \cdot (-3) = -4 - 8 + 9 = -3 \)
Thus, we find:
\[
A^2 = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} = A
\]
Since \( A^2 = A \), \( A \) is an idempotent matrix.
### Step 2: Check if \( A \) is a nilpotent matrix
A nilpotent matrix satisfies the condition \( A^p = 0 \) for some positive integer \( p \).
Since we already found \( A^2 = A \) and not the zero matrix, \( A \) is not nilpotent.
### Step 3: Check if \( A \) is an involutary matrix
An involutary matrix satisfies the condition \( A^2 = I \), where \( I \) is the identity matrix.
Since \( A^2 = A \) and not the identity matrix, \( A \) is not involutary.
### Step 4: Check if \( A \) is an orthogonal matrix
An orthogonal matrix satisfies the condition \( A^T A = I \).
Calculating \( A^T \):
\[
A^T = \begin{bmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{bmatrix}
\]
Now, compute \( A^T A \):
\[
A^T A = \begin{bmatrix} 2 & -1 & 1 \\ -2 & 3 & -2 \\ -4 & 4 & -3 \end{bmatrix} \cdot \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}
\]
Calculating the elements of \( A^T A \):
- First row, first column: \( 2 \cdot 2 + (-1) \cdot (-1) + 1 \cdot 1 = 4 + 1 + 1 = 6 \)
- First row, second column: \( 2 \cdot (-2) + (-1) \cdot 3 + 1 \cdot (-2) = -4 - 3 - 2 = -9 \)
- First row, third column: \( 2 \cdot (-4) + (-1) \cdot 4 + 1 \cdot (-3) = -8 - 4 - 3 = -15 \)
- Second row, first column: \( -2 \cdot 2 + 3 \cdot (-1) + (-2) \cdot 1 = -4 - 3 - 2 = -9 \)
- Second row, second column: \( -2 \cdot (-2) + 3 \cdot 3 + (-2) \cdot (-2) = 4 + 9 + 4 = 17 \)
- Second row, third column: \( -2 \cdot (-4) + 3 \cdot 4 + (-2) \cdot (-3) = 8 + 12 + 6 = 26 \)
- Third row, first column: \( -4 \cdot 2 + 4 \cdot (-1) + (-3) \cdot 1 = -8 - 4 - 3 = -15 \)
- Third row, second column: \( -4 \cdot (-2) + 4 \cdot 3 + (-3) \cdot (-2) = 8 + 12 + 6 = 26 \)
- Third row, third column: \( -4 \cdot (-4) + 4 \cdot 4 + (-3) \cdot (-3) = 16 + 16 + 9 = 41 \)
Thus, we find:
\[
A^T A = \begin{bmatrix} 6 & -9 & -15 \\ -9 & 17 & 26 \\ -15 & 26 & 41 \end{bmatrix}
\]
Since \( A^T A \neq I \), \( A \) is not orthogonal.
### Conclusion
The only property that holds true for the matrix \( A \) is that it is an **idempotent matrix**.
