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3 and 4 .Determinants and Matrices
normal
If $'a'$ is non real complex number for which system of equations $ax -a^2y + a^3z$ = $0$ , $-a^2x + a^3y + az$ = $0$ and $a^3x + ay -a^2z$ = $0$ has non trivial solutions, then $|a|$ is
A
$0$
B
$1$
C
$\sqrt3 $
D
$2$
Solution
$\left|\begin{array}{ccc}{a} & {-a^{2}} & {a^{3}} \\ {-a^{2}} & {a^{3}} & {a} \\ {a^{3}} & {a} & {-a^{2}}\end{array}\right|=0$
$\Rightarrow a^{3}(a+1)^{2}\left(a^{2}-a+1\right)^{2}=0$
$ \Rightarrow {\rm{a}} = 0, – 1, – \omega , – {\omega ^2}$
$\Rightarrow \mathrm{a}=-\omega,-\omega^{2}$ (non real)
Standard 12
Mathematics
