Find the multiplicative inverse of the following
`(3 +i)/(1 +i)`

Answer

To find the multiplicative inverse of the complex number \(\frac{3 + i}{1 + i}\), we will follow these steps:

### Step 1: Identify the complex number
The complex number we want to find the multiplicative inverse for is:
\[ z = \frac{3 + i}{1 + i} \]

### Step 2: Write the multiplicative inverse
The multiplicative inverse of a complex number \(z\) is given by:
\[ z^{-1} = \frac{1}{z} \]
Thus, we have:
\[ z^{-1} = \frac{1}{\frac{3 + i}{1 + i}} = \frac{1 + i}{3 + i} \]

### Step 3: Rationalize the denominator
To simplify \(\frac{1 + i}{3 + i}\), we will multiply the numerator and the denominator by the conjugate of the denominator, which is \(3 - i\):
\[ z^{-1} = \frac{(1 + i)(3 - i)}{(3 + i)(3 - i)} \]

### Step 4: Expand the numerator
Now, we expand the numerator:
\[ (1 + i)(3 - i) = 1 \cdot 3 + 1 \cdot (-i) + i \cdot 3 + i \cdot (-i) = 3 - i + 3i - i^2 \]
Since \(i^2 = -1\), we have:
\[ 3 - i + 3i + 1 = 3 + 1 + 2i = 4 + 2i \]

### Step 5: Expand the denominator
Next, we expand the denominator:
\[ (3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

### Step 6: Combine the results
Now we can combine the results from the numerator and denominator:
\[ z^{-1} = \frac{4 + 2i}{10} \]

### Step 7: Simplify the expression
We can simplify this by dividing both the real and imaginary parts by 10:
\[ z^{-1} = \frac{4}{10} + \frac{2i}{10} = \frac{2}{5} + \frac{i}{5} \]

### Final Answer
Thus, the multiplicative inverse of \(\frac{3 + i}{1 + i}\) is:
\[ \frac{2}{5} + \frac{i}{5} \]

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