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

Answer

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

### Step 1: Identify the complex number
We have \( z = 2 - 3i \).

### Step 2: Find the conjugate of \( z \)
The conjugate of a complex number \( a + bi \) is \( a - bi \). Therefore, the conjugate of \( z \) is:
\[ \overline{z} = 2 + 3i \]

### Step 3: Calculate the modulus of \( z \)
The modulus of a complex number \( z = a + bi \) is given by:
\[ |z| = \sqrt{a^2 + b^2} \]
For our \( z = 2 - 3i \):
\[ |z| = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 4: Calculate the multiplicative inverse
The multiplicative inverse \( z^{-1} \) of a complex number \( z \) can be calculated using the formula:
\[ z^{-1} = \frac{\overline{z}}{|z|^2} \]
Where \( |z|^2 \) is the square of the modulus. We already found \( \overline{z} = 2 + 3i \) and \( |z| = \sqrt{13} \), so:
\[ |z|^2 = (\sqrt{13})^2 = 13 \]
Now substituting these values into the formula:
\[ z^{-1} = \frac{2 + 3i}{13} \]

### Step 5: Write the result in standard form
We can express the multiplicative inverse as:
\[ z^{-1} = \frac{2}{13} + \frac{3}{13}i \]

Thus, the multiplicative inverse of \( z = 2 - 3i \) is:
\[ z^{-1} = \frac{2}{13} + \frac{3}{13}i \]

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