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

Options

A. `A^(3)-A^(2)=A-I`
B. `Det(A^(2010)-I)=0`
C. `A^(50)=[{:(1,0,0),(25,1,0),(25,0,1):}]`
D. `A^(50)=[{:(1,1,0),(25,1,0),(25,0,1):}]`

Explanation

To solve the problem, we need to find the adjoint and inverse of the matrix \( A \) given as:

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

### Step 1: Find the Determinant of Matrix A

To find the adjoint and inverse of a matrix, we first need to calculate its determinant.

\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 0 & 1 \\ 1 & 0 \end{vmatrix} - 0 + 0 \]

Calculating the 2x2 determinant:

\[ \begin{vmatrix} 0 & 1 \\ 1 & 0 \end{vmatrix} = (0 \cdot 0) - (1 \cdot 1) = -1 \]

Thus,

\[ \text{det}(A) = 1 \cdot (-1) = -1 \]

### Step 2: Find the Adjoint of Matrix A

The adjoint of a matrix is the transpose of its cofactor matrix. We will find the cofactor matrix first.

1. **Cofactor of \( a_{11} \)**:
\[ C_{11} = \begin{vmatrix} 0 & 1 \\ 1 & 0 \end{vmatrix} = -1 \]

2. **Cofactor of \( a_{12} \)**:
\[ C_{12} = -\begin{vmatrix} 1 & 1 \\ 0 & 0 \end{vmatrix} = 0 \]

3. **Cofactor of \( a_{13} \)**:
\[ C_{13} = \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = 1 \]

4. **Cofactor of \( a_{21} \)**:
\[ C_{21} = -\begin{vmatrix} 0 & 0 \\ 1 & 0 \end{vmatrix} = 0 \]

5. **Cofactor of \( a_{22} \)**:
\[ C_{22} = \begin{vmatrix} 1 & 0 \\ 0 & 0 \end{vmatrix} = 0 \]

6. **Cofactor of \( a_{23} \)**:
\[ C_{23} = -\begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = -1 \]

7. **Cofactor of \( a_{31} \)**:
\[ C_{31} = \begin{vmatrix} 0 & 0 \\ 1 & 0 \end{vmatrix} = 0 \]

8. **Cofactor of \( a_{32} \)**:
\[ C_{32} = -\begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix} = -1 \]

9. **Cofactor of \( a_{33} \)**:
\[ C_{33} = \begin{vmatrix} 1 & 0 \\ 1 & 0 \end{vmatrix} = 0 \]

Now, we can construct the cofactor matrix:

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

Taking the transpose to get the adjoint:

\[ \text{adj}(A) = C^T = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix} \]

### Step 3: Find the Inverse of Matrix A

The inverse of a matrix \( A \) can be found using the formula:

\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]

Since \( \text{det}(A) = -1 \):

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

### Final Result

The adjoint of matrix \( A \) is:

\[ \text{adj}(A) = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix} \]