If `A=[[2,-2,-4],[-1,3,4],[1,-2,-3]]` then A is `1) an idempotent matrix 2) nilpotent matrix 3) involutary 4) orthogonal matrix

Explanation

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, involutory matrix, or orthogonal matrix.

### Step 1: Check if \( A \) is an Idempotent Matrix
A matrix \( A \) is idempotent if \( A^2 = A \).

**Calculation of \( A^2 \):**
\[
A^2 = A \times A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix} \times \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}
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