If A = `[{:( 0,0,0),( 1,0,0),(0,1,0):}]` ,then

Options

A. `A^(2) =A `
B. ` A^(2) = 0`
C. ` A^(2)= I`
D. `A^(3) = O`

Answer

To solve the problem, we need to find \( A^2 \) and \( A^3 \) for the given matrix \( A \).

Given:
\[ A = \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \]

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

To find \( A^2 \), we perform matrix multiplication \( A \cdot A \).

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

Calculating each element of \( A^2 \):

- First row:
- First element: \( 0 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0 \)
- Second element: \( 0 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 = 0 \)
- Third element: \( 0 \cdot 0 + 0 \cdot 0 + 0 \cdot 0 = 0 \)

- Second row:
- First element: \( 1 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0 \)
- Second element: \( 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 = 0 \)
- Third element: \( 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 0 = 0 \)

- Third row:
- First element: \( 0 \cdot 0 + 1 \cdot 1 + 0 \cdot 0 = 1 \)
- Second element: \( 0 \cdot 0 + 1 \cdot 0 + 0 \cdot 1 = 0 \)
- Third element: \( 0 \cdot 0 + 1 \cdot 0 + 0 \cdot 0 = 0 \)

Thus, we have:
\[ A^2 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \]

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

Now, we need to find \( A^3 \) by multiplying \( A^2 \) with \( A \).

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

Calculating each element of \( A^3 \):

- First row:
- First element: \( 0 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0 \)
- Second element: \( 0 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 = 0 \)
- Third element: \( 0 \cdot 0 + 0 \cdot 0 + 0 \cdot 0 = 0 \)

- Second row:
- First element: \( 0 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0 \)
- Second element: \( 0 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 = 0 \)
- Third element: \( 0 \cdot 0 + 0 \cdot 0 + 0 \cdot 0 = 0 \)

- Third row:
- First element: \( 1 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0 \)
- Second element: \( 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 = 0 \)
- Third element: \( 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 0 = 0 \)

Thus, we have:
\[ A^3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

### Final Results

1. \( A^2 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \)
2. \( A^3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \)