If z is a complex number, then `( bar( z) ^(-1)) ( bar( z))` is equal to

Options

A. 1
B. `-1`
C. 0
D. None of these

Answer

To solve the problem, we need to evaluate the expression \((\bar{z})^{-1} \cdot \bar{z}\), where \(z\) is a complex number and \(\bar{z}\) is its conjugate.

### Step-by-step Solution:

1. **Define the Complex Number**:
Let \(z = x + iy\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit.

2. **Find the Conjugate**:
The conjugate of \(z\) is given by:
\[ \bar{z} = x - iy \]

3. **Find the Inverse of the Conjugate**:
The inverse of \(\bar{z}\) is:
\[ (\bar{z})^{-1} = \frac{1}{\bar{z}} = \frac{1}{x - iy} \]
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ (\bar{z})^{-1} = \frac{1}{x - iy} \cdot \frac{x + iy}{x + iy} = \frac{x + iy}{x^2 + y^2} \]

4. **Multiply by the Conjugate**:
Now we need to compute:
\[ (\bar{z})^{-1} \cdot \bar{z} = \left(\frac{x + iy}{x^2 + y^2}\right) \cdot (x - iy) \]
This simplifies to:
\[ = \frac{(x + iy)(x - iy)}{x^2 + y^2} \]

5. **Expand the Numerator**:
Using the difference of squares:
\[ (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 + y^2 \]

6. **Combine the Results**:
Now substituting back into our expression:
\[ (\bar{z})^{-1} \cdot \bar{z} = \frac{x^2 + y^2}{x^2 + y^2} = 1 \]

### Final Answer:
Thus, the expression \((\bar{z})^{-1} \cdot \bar{z}\) is equal to \(1\).

---