The number of `2x2` matrices `X` satisfying the matrix equation `X^2=I(Ii s2x2u n i tm a t r i x)` is 1 (b) 2 (c) 3 (d) infinite

Options

A. 0
B. 1
C. 2
D. more then 3

Explanation

To solve the problem, we need to find the number of \(2 \times 2\) matrices \(X\) that satisfy the equation \(X^2 = I\), where \(I\) is the \(2 \times 2\) identity matrix.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \(X^2 = I\) implies that multiplying the matrix \(X\) by itself gives the identity matrix. This means that \(X\) is its own inverse, i.e., \(X = X^{-1}\).

**Hint**: Recall that the identity matrix \(I\) has the property that \(I \cdot A = A \cdot I = A\) for any matrix \(A\).

2. **Multiplying by \(X^{-1}\)**:
If we multiply both sides of the equation \(X^2 = I\) by \(X^{-1}\) (the inverse of \(X\)), we get:
\[ X \cdot X = I \implies X = X^{-1} \]

**Hint**: Remember that multiplying both sides of an equation by the same non-zero matrix does not change the equality.

3. **Characterizing Matrices**:
The condition \(X = X^{-1}\) means that \(X\) is an involutory matrix. In \(2 \times 2\) matrices, this can be satisfied by matrices that have eigenvalues of either \(1\) or \(-1\).

**Hint**: Consider the properties of eigenvalues and how they relate to the matrix being its own inverse.

4. **Finding Possible Matrices**:
A general form of a \(2 \times 2\) matrix is:
\[ X = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
For \(X\) to satisfy \(X^2 = I\), we can derive conditions on the entries \(a\), \(b\), \(c\), and \(d\). The characteristic polynomial of \(X\) must be \(\lambda^2 - 1 = 0\), leading to eigenvalues of \(1\) and \(-1\).

**Hint**: Think about the implications of the eigenvalues on the structure of the matrix.

5. **Constructing Examples**:
Some specific examples of \(2 \times 2\) matrices that satisfy \(X^2 = I\) include:
- The identity matrix \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
- The matrix \(X_1 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
- The matrix \(X_2 = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
- The matrix \(X_3 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)

**Hint**: Try to find other matrices that can be represented in a similar form.

6. **Conclusion**:
Since there are infinitely many matrices that can be constructed to satisfy the condition \(X = X^{-1}\) (for example, any rotation or reflection matrix), we conclude that there are infinitely many \(2 \times 2\) matrices \(X\) such that \(X^2 = I\).

**Final Answer**: The number of \(2 \times 2\) matrices \(X\) satisfying \(X^2 = I\) is (d) infinite.