Consider the following in respect of a complex number Z :
1. `bar((Z^(-1)))=(barZ)^(-1)`
2. `Z Z^(-1) =|Z|^(2)`
Which of the above is/are correct?

Options

A. `1` only
B. `2` only
C. Both `1` and `2`
D. Neither `1` nor `2`

Answer

To solve the problem, we need to analyze the two statements regarding the complex number \( Z \).

### Step 1: Analyze the first statement
The first statement is:
\[ \overline{(Z^{-1})} = (\overline{Z})^{-1} \]

**Solution:**
1. Let \( Z = a + i b \), where \( a \) and \( b \) are real numbers.
2. The conjugate of \( Z \) is:
\[ \overline{Z} = a - i b \]
3. The inverse of \( Z \) is given by:
\[ Z^{-1} = \frac{\overline{Z}}{|Z|^2} = \frac{a - i b}{a^2 + b^2} \]
4. Now, find the conjugate of \( Z^{-1} \):
\[ \overline{(Z^{-1})} = \overline{\left(\frac{a - i b}{a^2 + b^2}\right)} = \frac{\overline{(a - i b)}}{|Z|^2} = \frac{a + i b}{a^2 + b^2} \]
5. Now, calculate \( (\overline{Z})^{-1} \):
\[ (\overline{Z})^{-1} = \frac{Z}{|Z|^2} = \frac{a + i b}{a^2 + b^2} \]
6. Since both expressions are equal:
\[ \overline{(Z^{-1})} = (\overline{Z})^{-1} \]
This statement is **true**.

### Step 2: Analyze the second statement
The second statement is:
\[ Z Z^{-1} = |Z|^2 \]

**Solution:**
1. We already know that \( Z^{-1} = \frac{\overline{Z}}{|Z|^2} \).
2. Now, calculate \( Z Z^{-1} \):
\[ Z Z^{-1} = Z \cdot \frac{\overline{Z}}{|Z|^2} = \frac{Z \overline{Z}}{|Z|^2} \]
3. Since \( Z \overline{Z} = |Z|^2 \), we have:
\[ Z Z^{-1} = \frac{|Z|^2}{|Z|^2} = 1 \]
4. Therefore, the statement \( Z Z^{-1} = |Z|^2 \) is **false** because \( Z Z^{-1} = 1 \).

### Conclusion
- The first statement is true.
- The second statement is false.

Thus, the correct answer is that only the first statement is correct.