Question

Let $A=\left[\begin{array}{ccc}1& a^3& 0\\ 0& 1& b^3\\ c^3& 0& 1\end{array}\right]$ where $a,b,c\in R$ . If the sum of all non-real roots of the equation $\left|A-xI\right|=0$ is $k-mabc,\forall k,m\in Z$ , then the value of $k+m$ is equal to

Answer 1

Answer: 3

$\left|A-nI\right|=\left|\begin{array}{ccc}1-n& a^3& 0\\ 0& 1-n& b\\ c^3& 0& 1-n\end{array}\right|=0$
$\Rightarrow \left(1-n\right)\left[{\left(1-n\right)}^2\right]-a^3\left[-b^3{c}^3\right]=0$
$\Rightarrow -{\left(n-1\right)}^3+a^3{b}^3{c}^3=0$
$\Rightarrow {\left(n-1\right)}^3={\left(abc\right)}^3$
$\Rightarrow n=1+abc,1+abcw,1+abc{w}^2$
Sum of non real roots $=2+abc\left(w+w^2\right)$
$=2-abc$
$k=2,m=1$
$\Rightarrow k+m=3$