Question

If $p$ and $q$ are distinct primes, and
$\Delta =\left|\begin{array}{ccc}\sqrt{pq}& pi& q+\sqrt{pq}\\ p\sqrt{q}& \sqrt{p}+p\sqrt{p}i& q\sqrt{p}+\sqrt{p}qi\\ q\sqrt{p}& \sqrt{p}+p\sqrt{q}i& q\sqrt{q}+pi\end{array}\right|$
then $\Delta$ is

• A. an integer
• B. irrational
• C. non-zerorational
• D. imaginary

Answer 1

The correct option is A an integer

$\Delta =\left|\begin{array}{ccc}\sqrt{pq}& pi& q+\sqrt{pq}\\ p\sqrt{q}& \sqrt{p}+p\sqrt{p}i& q\sqrt{p}+\sqrt{p}qi\\ q\sqrt{p}& \sqrt{p}+p\sqrt{q}i& q\sqrt{q}+pi\end{array}\right|$

$=\left|\begin{array}{ccc}\sqrt{pq}& 0& q\\ p\sqrt{q}& \sqrt{p}& q\sqrt{p}\\ q\sqrt{p}& \sqrt{p}& q\sqrt{q}\end{array}\right|+\left|\begin{array}{ccc}0& pi& \sqrt{pq}\\ 0& p\sqrt{p}i& \sqrt{p}qi\\ 0& p\sqrt{q}i& pi\end{array}\right|$

$=\left(\sqrt{pq}\right)\left(\sqrt{p}\right)\left(q\right)\left|\begin{array}{ccc}1& 0& 1\\ \sqrt{p}& 1& \sqrt{p}\\ \sqrt{q}& 1& \sqrt{q}\end{array}\right|$

$=\left(\sqrt{pq}\right)\left(\sqrt{p}\right)\left(q\right)(1(\sqrt{pq}-\sqrt{pq})+1(\sqrt{pq}-\sqrt{pq}))=0$