Question

Find the multiplicative inverse of 12 – 5i.
1. \(\frac{{12}}{{13}} + \frac{5}{{13}}i\)
2. 12 + 5i
3. \(\frac{{12}}{{169}} - \frac{5}{{169}}i\)
4. \(\frac{{12}}{{169}} + \frac{5}{{169}}i\)

Answer 1

Correct Answer - Option 4 : \(\frac{{12}}{{169}} + \frac{5}{{169}}i\)

CONCEPT:

Let z = a + ib be a complex number. Then, the modulus of z, denoted by |z|, is defined to be the non-negative real number \(\sqrt {{a^2} + {b^2}} .\)

And the conjugate of z, denoted as z̅,  is the complex number a – ib,

Multiplicative inverse of the non-zero complex number z is given by \({z^{ - 1}} = \frac{\bar z}{{{{\left| z \right|}^2}}}\)

CALCULATION:

Given complex number is 12 – 5i

∴ z̅ = 12 + 5i and |z|² = (12)² + (-5)² = 169

Therefore, the multiplicative inverse of 12 – 5i is given by –

\({z^{ - 1}} = \frac{{̅ z}}{{{{\left| z \right|}^2}}} = \frac{{12 + 5{\rm{i}}}}{{169}} = \frac{{12}}{{169}} + \frac{5}{{169}}i\)