Question

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

Answer 1

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