Question

$LetP=\left[\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right],abc=1anda,b,c,ϵRandX={\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]}_{3\times 1};thenPX=0$
has infinitely many solution if trace (P) is _________.

• A. 3

Answer 1

The correct option is A 3
$\because PX=0hasinfinitelynumbersolutions$
$\therefore |P|=0$
$i.e.det\left(P\right)=\left[\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right]=0$

$\Rightarrow a(bc-1)-1(c-1)+1(1-b)=0$
$\Rightarrow abc-a-c+1+1-b=0$
$\Rightarrow 1-a-b-c+2=0\{\because abc=1\}$
$\Rightarrow a+b+c=3$

For matrix (P)

trace (P) = sum of diagonal elements
tr(P) = a +b + c
tr (P) = 3