Find the multiplicative inverser of the following
`(-2,1)`

Answer

To find the multiplicative inverse of the complex number \((-2, 1)\), we will follow these steps:

### Step 1: Write the complex number in standard form
The complex number \((-2, 1)\) can be expressed in standard form as:
\[ z = -2 + i \]

### Step 2: Find the multiplicative inverse
The multiplicative inverse of a complex number \(z\) is given by:
\[ z^{-1} = \frac{1}{z} \]
For our complex number, this becomes:
\[ z^{-1} = \frac{1}{-2 + i} \]

### Step 3: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ z^{-1} = \frac{1}{-2 + i} \cdot \frac{-2 - i}{-2 - i} \]
This simplifies to:
\[ z^{-1} = \frac{-2 - i}{(-2 + i)(-2 - i)} \]

### Step 4: Calculate the denominator
Now we calculate the denominator:
\[ (-2 + i)(-2 - i) = (-2)^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5 \]

### Step 5: Substitute the denominator back into the equation
Now we can substitute the calculated denominator back into our expression for the multiplicative inverse:
\[ z^{-1} = \frac{-2 - i}{5} \]

### Step 6: Write the final answer
Thus, the multiplicative inverse of the complex number \((-2, 1)\) is:
\[ z^{-1} = -\frac{2}{5} - \frac{1}{5}i \]

### Summary
The multiplicative inverse of the complex number \((-2, 1)\) is:
\[ -\frac{2}{5} - \frac{1}{5}i \]