Let A be a `2xx2` matrix with real entries. Let I be the `2xx2` identity matrix. Denoted by tr(A), the sum of diagonal entries of A. Assume that `A^(2)=I`
Statement 1: If `A!=I` and `A=-I`, then det A=-1
Statement 2: If `A!=I` and A=-I, then tr `A!=0`.

Answer

To solve the problem, we will analyze the statements given about the matrix \( A \) under the condition \( A^2 = I \) where \( I \) is the identity matrix.

### Step 1: Understanding the condition \( A^2 = I \)

Given that \( A^2 = I \), we can infer that \( A \) is an involutory matrix. This means that multiplying the matrix by itself gives the identity matrix.

### Step 2: Matrix representation

Let \( A \) be represented as:
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
The identity matrix \( I \) is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 3: Setting up the equation

From \( A^2 = I \), we can write:
\[ A \cdot A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

This gives us the following equations:
1. \( a^2 + bc = 1 \)
2. \( ab + bd = 0 \)
3. \( ac + cd = 0 \)
4. \( bc + d^2 = 1 \)

### Step 4: Analyzing Statement 1

**Statement 1:** If \( A \neq I \) and \( A = -I \), then \( \det A = -1 \).

If \( A = -I \), then:
\[ A = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]
The determinant of \( A \) is:
\[ \det A = (-1)(-1) - (0)(0) = 1 \]
Thus, the statement is incorrect because \( \det A \) is not equal to \(-1\), it is \(1\).

### Step 5: Analyzing Statement 2

**Statement 2:** If \( A \neq I \) and \( A = -I \), then \( \text{tr}(A) \neq 0 \).

The trace of \( A \) is given by:
\[ \text{tr}(A) = a + d \]
For \( A = -I \):
\[ \text{tr}(A) = -1 + (-1) = -2 \neq 0 \]
Thus, this statement is true.

### Conclusion

- **Statement 1** is **false** because \( \det A = 1 \) when \( A = -I \).
- **Statement 2** is **true** because \( \text{tr}(A) = -2 \neq 0 \).