If `A=[(x,1),(1,0)]` and `A^(2)` is the identity matrix, then `x` is equal to

Options

A. `-1`
B. `0`
C. `1`
D. `2`

Explanation

To solve the problem, we need to find the value of \( x \) such that the matrix \( A = \begin{pmatrix} x & 1 \\ 1 & 0 \end{pmatrix} \) satisfies the condition \( A^2 = I \), where \( I \) is the identity matrix.

### Step-by-Step Solution:

1. **Calculate \( A^2 \)**:
\[ A^2 = A \cdot A = \begin{pmatrix} x & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} x & 1 \\ 1 & 0 \end{pmatrix} \]
To multiply the matrices, we use the formula for matrix multiplication:
\[ A^2 = \begin{pmatrix} x \cdot x + 1 \cdot 1 & x \cdot 1 + 1 \cdot 0 \\ 1 \cdot x + 0 \cdot 1 & 1 \cdot 1 + 0 \cdot 0 \end{pmatrix} \]
Simplifying this, we get:
\[ A^2 = \begin{pmatrix} x^2 + 1 & x \\ x & 1 \end{pmatrix} \]

2. **Set \( A^2 \) equal to the identity matrix**:
The identity matrix \( I \) for a 2x2 matrix is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
Therefore, we set:
\[ \begin{pmatrix} x^2 + 1 & x \\ x & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

3. **Equate the corresponding elements**:
From the equality of matrices, we can derive the following equations:
- From the first element: \( x^2 + 1 = 1 \)
- From the second element: \( x = 0 \)
- From the third element: \( x = 0 \)
- From the fourth element: \( 1 = 1 \) (which is always true)

4. **Solve the first equation**:
From \( x^2 + 1 = 1 \):
\[ x^2 = 1 - 1 = 0 \]
Taking the square root of both sides gives:
\[ x = 0 \]

5. **Conclusion**:
The value of \( x \) that satisfies the condition \( A^2 = I \) is:
\[ \boxed{0} \]