Question

If $z_{1}=1+3i$, $z_{2}=-3i+5$, then $(z_{1}\overline{z_{2}}+z_{2}\overline{z_{1}})+(z_{1}\overline{z_{2}}+z_{2}\overline{z_{1}})$ is equal to

• A. -16
• B. 1+i
• C. 1
• D. 1-i
• E. -16i

Hint:

Math Tip: The expression $z\bar{w} + \bar{z}w$ is always equal to $2\text{Re}(z\bar{w})$. Knowing this shortcut lets you skip evaluating the second product completely. Just calculate the real part of the first product and double it!

Answer 1

The Correct Option is A

Concept:
Complex Numbers - Multiplication of a Complex Number and a Conjugate.
Step 1: Identify the values and their conjugates.
Given $z_1 = 1 + 3i$, its conjugate is $\overline{z_1} = 1 - 3i$.
Given $z_2 = 5 - 3i$ (rewriting $-3i+5$ in standard form), its conjugate is $\overline{z_2} = 5 + 3i$.
Step 2: Evaluate the term $z_1\overline{z_2}$.
Multiply $z_1$ and $\overline{z_2}$: $$ z_1\overline{z_2} = (1 + 3i)(5 + 3i) $$ $$ = 5 + 3i + 15i + 9i^2 $$ Substitute $i^2 = -1$: $$ = 5 + 18i - 9 $$ $$ z_1\overline{z_2} = -4 + 18i $$
Step 3: Evaluate the term $z_2\overline{z_1}$.
Multiply $z_2$ and $\overline{z_1}$: $$ z_2\overline{z_1} = (5 - 3i)(1 - 3i) $$ $$ = 5 - 15i - 3i + 9i^2 $$ Substitute $i^2 = -1$: $$ = 5 - 18i - 9 $$ $$ z_2\overline{z_1} = -4 - 18i $$
Step 4: Sum the evaluated terms.
Add the two results together: $$ z_1\overline{z_2} + z_2\overline{z_1} = (-4 + 18i) + (-4 - 18i) $$ The imaginary parts cancel out: $$ z_1\overline{z_2} + z_2\overline{z_1} = -8 $$
Step 5: Evaluate the full given expression.
The original expression asks for $(z_1\overline{z_2} + z_2\overline{z_1}) + (z_1\overline{z_2} + z_2\overline{z_1})$.
Substitute the sum we just found: $$ (-8) + (-8) = -16 $$