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

Options

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

Answer

To solve the problem, we need to find \( A^2 \) where \( A \) is given as:

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

### Step 1: Write down the matrix A
We have the matrix \( A \):

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

### Step 2: Calculate \( A^2 \) (which is \( A \times A \))
To find \( A^2 \), we perform the matrix multiplication \( A \times A \).

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

### Step 3: Perform the multiplication
We will calculate each element of the resulting matrix \( A^2 \).

- **Element (1,1)**:
\[ 0 \times 0 + 0 \times 0 + 0 \times 1 = 0 \]

- **Element (1,2)**:
\[ 0 \times 0 + 0 \times 0 + 0 \times 0 = 0 \]

- **Element (1,3)**:
\[ 0 \times 0 + 0 \times 0 + 0 \times 0 = 0 \]

- **Element (2,1)**:
\[ 0 \times 0 + 0 \times 0 + 0 \times 1 = 0 \]

- **Element (2,2)**:
\[ 0 \times 0 + 0 \times 0 + 0 \times 0 = 0 \]

- **Element (2,3)**:
\[ 0 \times 0 + 0 \times 0 + 0 \times 0 = 0 \]

- **Element (3,1)**:
\[ 1 \times 0 + 0 \times 0 + 0 \times 1 = 0 \]

- **Element (3,2)**:
\[ 1 \times 0 + 0 \times 0 + 0 \times 0 = 0 \]

- **Element (3,3)**:
\[ 1 \times 0 + 0 \times 0 + 0 \times 0 = 0 \]

### Step 4: Write down the resulting matrix
After performing the multiplication, we find:

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

### Conclusion
Thus, we conclude that:

\[ A^2 = 0 \]