Question

Prove that the value of the determinant $\left[\begin{array}{ccc}-7& 5+3i& \frac{2}{3}-4i\\ 5-3i& 8& 4+5i\\ \frac{2}{3}+4i& 4-5i& 9\end{array}\right]$ is real.

Answer 1

$\begin{array}{c}\left|A\right|=-7\left|\begin{array}{cc}8& 4+5i\\ 4-5i& 9\end{array}\right|-5+3i\left|\begin{array}{cc}5-3i& 4+5i\\ \frac{2}{3}+4i& 9\end{array}\right|+\left(\frac{2}{3}-4i\right)\left|\begin{array}{cc}5-3i& 8\\ \frac{2}{3}+4i& 4-5i\end{array}\right|\\ =-7\left[72-\left\{16+25\right\}\right]-\left(5+3i\right)\left\{9\left(5-3i\right)-\left(4+5i\right)\left(\frac{2}{3}+4i\right)+\left(\frac{2}{3}-4i\right)\left(5-3i\right)\left(4-5i\right)-8\left(\frac{2}{3}+40\right)\right\}\end{array}$

If after product i part cancelled

only Real value Remain.