If A = `[ (-5,-8,0),(3,5,0),(1,2,-1)]` then A is

Options

A. idempotent
B. nilpotent
C. involutory
D. None

Answer

To determine the nature of the matrix \( A \) given by

\[ A = \begin{pmatrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{pmatrix} \]

we will first compute \( A^2 \) and check if it is the zero matrix. If \( A^2 = 0 \), then \( A \) is a nilpotent matrix.

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{pmatrix} \cdot \begin{pmatrix} -5 & -8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & -1 \end{pmatrix} \]

### Step 2: Perform the multiplication

We will calculate each element of \( A^2 \):

- **First row, first column**:
\[ (-5)(-5) + (-8)(3) + (0)(1) = 25 - 24 + 0 = 1 \]

- **First row, second column**:
\[ (-5)(-8) + (-8)(5) + (0)(2) = 40 - 40 + 0 = 0 \]

- **First row, third column**:
\[ (-5)(0) + (-8)(0) + (0)(-1) = 0 + 0 + 0 = 0 \]

- **Second row, first column**:
\[ (3)(-5) + (5)(3) + (0)(1) = -15 + 15 + 0 = 0 \]

- **Second row, second column**:
\[ (3)(-8) + (5)(5) + (0)(2) = -24 + 25 + 0 = 1 \]

- **Second row, third column**:
\[ (3)(0) + (5)(0) + (0)(-1) = 0 + 0 + 0 = 0 \]

- **Third row, first column**:
\[ (1)(-5) + (2)(3) + (-1)(1) = -5 + 6 - 1 = 0 \]

- **Third row, second column**:
\[ (1)(-8) + (2)(5) + (-1)(2) = -8 + 10 - 2 = 0 \]

- **Third row, third column**:
\[ (1)(0) + (2)(0) + (-1)(-1) = 0 + 0 + 1 = 1 \]

### Step 3: Assemble \( A^2 \)

Combining all the calculated elements, we get:

\[ A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Conclusion

Since \( A^2 \) is not the zero matrix, we conclude that \( A \) is not a nilpotent matrix.

### Final Answer

The matrix \( A \) is **not nilpotent**.

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