Question

The value of $\left|\begin{array}{ccc}{i}^m& i^{m+1}& i^{m+2}\\ i^{m+5}& i^{m+4}& i^{m+3}\\ i^{m+6}& i^{m+7}& i^{m+8}\end{array}\right|,$ where $i=\sqrt{-1},$ is

• A. $1$ if $m$ is a multiple of $4$
• B. $0$ for all real $m$
• C. $-i$ if $m$ is a multiple of $3$
• D. none of these

Answer 1

Answer: (B)

$0$ for all real $m$

$\left|\begin{array}{ccc}{i}^m& i^{m+1}& i^{m+2}\\ i^{m+5}& i^{m+4}& i^{m+3}\\ i^{m+6}& i^{m+7}& i^{m+8}\end{array}\right|$
$=i^m{i}^{m+1}{i}^{m+2}\left|\begin{array}{ccc}1& 1& 1\\ i^5& i^3& i\\ i^6& i^6& i^6\end{array}\right|$
$=i^{3m+3}{i}^6\left|\begin{array}{ccc}1& 1& 1\\ i^5& i^3& i\\ 1& 1& 1\end{array}\right|$
$=0$
Hence, for all real m , the value of given determinant is 0.