Question

If $i=\sqrt{-1}$, then find the value of the determinant $\left|\begin{array}{ccc}{\sum }_{m=1}^{4n}{i}^m& {\sum }_{m=1}^{4n+1}{i}^{m+1}& {\sum }_{m=1}^{4n+2}{i}^{m+2}\\ {\sum }_{m=1}^{4n+1}{i}^m& {\sum }_{m=1}^{4n+2}{i}^{m+1}& {\sum }_{m=1}^{4n+3}{i}^{m+2}\\ {\sum }_{m=1}^{4n+2}{i}^m& {\sum }_{m=1}^{4n+3}{i}^{m+1}& {\sum }_{m=1}^{4n+4}{i}^{m+2}\end{array}\right|$

Answer 1

$\left|\begin{array}{ccc}{\sum }_{m=1}^{4n}{i}^m& {\sum }_{m=1}^{4n+1}{i}^{m+1}& {\sum }_{m=1}^{4n+2}{i}^{m+2}\\ {\sum }_{m=1}^{4n+1}{i}^m& {\sum }_{m=1}^{4n+2}{i}^{m+1}& {\sum }_{m=1}^{4n+3}{i}^{m+2}\\ {\sum }_{m=1}^{4n+2}{i}^m& {\sum }_{m=1}^{4n+3}{i}^{m+1}& {\sum }_{m=1}^{4n+4}{i}^{m+2}\end{array}\right|$
$=\left|\begin{array}{ccc}i+i^2+...+i^{4n}& i^2+i^3+...+i^{4n}+i^{4n+1}+i^{4n+2}& i^3+i^4+...+i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}+i^{4n+4}\\ i+i^2+...+i^{4n}+i^{4n+1}& i^2+i^3+...+i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}& i^3+i^4+...+i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}+i^{4n+4}+i^{4n+5}\\ i+i^2+...+i^{4n}+i^{4n+1}+i^{4n+2}& i^2+i^3+...+i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}+i^{4n+4}& i^3+i^4+...+i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}+i^{4n+4}+i^{4n+5}+i^{4n+6}\end{array}\right|$
$R_3\to R_3-R_2,R_2\to R_2-R_1$
$=\left|\begin{array}{ccc}i+i^2+...+i^{4n}& i^2+i^3+...+i^{4n}+i^{4n+1}+i^{4n+2}& i^3+i^4+...+i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}+i^{4n+4}\\ i^{4n+1}& i^{4n+3}& i^{4n+5}\\ i^{4n+2}& i^{4n+4}& i^{4n+6}\end{array}\right|$
$=\left|\begin{array}{ccc}i+i^2+...+i^{4n}& i^2+i^3+...+i^{4n}+i^{4n+1}+i^{4n+2}& i^3+i^4+...+i^{4n}+i^{4n+1}+i^{4n+2}+i^{4n+3}+i^{4n+4}\\ i& -i& i\\ -1& 1& -1\end{array}\right|$
$=\left|\begin{array}{ccc}i+i^2+...+i^{4n}& -i+i^{4n+1}+i^{4n+2}& -i^2+i^{4n+3}+i^{4n+4}\\ i& -2i& 2i\\ -1& 2& -2\end{array}\right|$
$=\left|\begin{array}{ccc}\frac{1-i^{4n}}{1-i}& -1& -i+2\\ i& -2i& 2i\\ -1& 2& -2\end{array}\right|$ ($a_{11}$ is a G.P. , $i^{4n}=1$)
$=\left|\begin{array}{ccc}0& -1& -i+2\\ i& -2i& 2i\\ -1& 2& -2\end{array}\right|$
$=0$