Question

If $abc=p$ and $A=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]$, if and only if $a,b,c$ are the roots of equation $x^3\pm x^2+p=0$.
then find the value of $det\left(A{A}^{\prime }\right)$

Answer 1

Given $abc=p$ and $A=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]$,if and only if $a,b,c$ are the roots of equation $x^3\pm x^2+p=0$

$\Rightarrow a+b+c=\pm 1,ab+bc+ca=0$ and $abc=p$

$a^2+b^2+c^2=(a+b+c{)}^2-2(ab+bc+ca)=1$

$A{A}^{\prime }=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]\left[\begin{array}{ccc}a& c& b\\ b& a& c\\ c& b& a\end{array}\right]=\left[\begin{array}{ccc}{a}^2+b^2+c^2& ab+bc+ca& ab+bc+ca\\ ab+bc+ca& a^2+b^2+c^2& ab+bc+ca\\ ab+bc+ca& ab+bc+ca& a^2+b^2+c^2\end{array}\right]$

$\Rightarrow A{A}^{\prime }=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I$

$\therefore |A{A}^{\prime }|=|I|=1$