Find the multiplicative inverse of `z=3-2i`.

Options

A. `(1+2i)/(5)`
B. `(1-2i)/(5)`
C. `(1+2i)/(sqrt(5))`
D. None of these

Answer

To find the multiplicative inverse of the complex number \( z = 3 - 2i \), we follow these steps:

### Step 1: Write the expression for the multiplicative inverse
The multiplicative inverse of \( z \) is given by \( \frac{1}{z} \). Therefore, we need to calculate:
\[ \frac{1}{3 - 2i} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3 - 2i \) is \( 3 + 2i \). Thus, we have:
\[ \frac{1}{3 - 2i} \cdot \frac{3 + 2i}{3 + 2i} = \frac{3 + 2i}{(3 - 2i)(3 + 2i)} \]

### Step 3: Calculate the denominator
Now, we need to calculate the denominator:
\[ (3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - (4)(-1) = 9 + 4 = 13 \]

### Step 4: Substitute back into the expression
Now we can substitute back into our expression:
\[ \frac{3 + 2i}{13} \]

### Step 5: Separate the real and imaginary parts
This can be expressed as:
\[ \frac{3}{13} + \frac{2}{13}i \]

### Conclusion
Thus, the multiplicative inverse of \( z = 3 - 2i \) is:
\[ \frac{3}{13} + \frac{2}{13}i \]

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