Question

If \( z = \frac{3+i}{2-i} \), then \( z^{-1} \) is equal to

• A. \(\frac{1+i}{2}\)
• B. \(\frac{1+i}{2}\)
• C. \(\frac{1-i}{2}\)
• D. \(2(1-i)\)
• E. \(1-i\)

Hint:

To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. Also, remember that \(z^{-1}=\frac{1}{z}\).

Answer 1

The Correct Option is C

Step 1: Write the given complex number clearly.
We are given:
\[ z=\frac{3+i}{2-i} \] We have to find \(z^{-1}\), which means the reciprocal of \(z\).

Step 2: Use the reciprocal property.
If \[ z=\frac{3+i}{2-i}, \] then \[ z^{-1}=\frac{1}{z}=\frac{2-i}{3+i} \] So now we only need to simplify:
\[ \frac{2-i}{3+i} \]

Step 3: Rationalize the denominator.
To remove the imaginary term from the denominator, multiply numerator and denominator by the conjugate of \(3+i\), which is \(3-i\):
\[ z^{-1}=\frac{2-i}{3+i}\cdot\frac{3-i}{3-i} \]

Step 4: Expand the numerator.
Now simplify the numerator:
\[ (2-i)(3-i)=2\cdot 3+2(-i)+(-i)3+(-i)(-i) \] \[ =6-2i-3i+i^2 \] Since \(i^2=-1\), this becomes:
\[ 6-5i-1=5-5i \] So the numerator is:
\[ 5-5i \]

Step 5: Expand the denominator.
Now simplify the denominator:
\[ (3+i)(3-i)=3^2-i^2=9-(-1)=10 \] Hence,
\[ z^{-1}=\frac{5-5i}{10} \]

Step 6: Reduce the expression.
Factor out \(5\) from the numerator:
\[ z^{-1}=\frac{5(1-i)}{10}=\frac{1-i}{2} \]

Step 7: Match with the given options.
Thus,
\[ z^{-1}=\frac{1-i}{2} \] This matches option \((3)\).