Question

$\left|\begin{array}{ccc}{a}^2+x^2& ab-cx& ac+bx\\ ab+cx& b^2+x^2& bc-ax\\ ac-bx& bc+ax& c^2+x^2\end{array}\right|=$

• A. ${\left|\begin{array}{ccc}x& b& -c\\ -a& x& c\\ a& -b& x\end{array}\right|}^2$
• B. ${\left|\begin{array}{ccc}x& -b& c\\ a& x& -c\\ -a& b& x\end{array}\right|}^2$
• C. ${\left|\begin{array}{ccc}x& c& -b\\ -c& x& a\\ b& -a& x\end{array}\right|}^2$
• D. ${\left|\begin{array}{ccc}x& -c& b\\ c& x& -a\\ -b& a& x\end{array}\right|}^2$

Answer 1

Answer: (C), (D)

The correct options are
C ${\left|\begin{array}{ccc}x& -c& b\\ c& x& -a\\ -b& a& x\end{array}\right|}^2$
D ${\left|\begin{array}{ccc}x& c& -b\\ -c& x& a\\ b& -a& x\end{array}\right|}^2$

Let $D=\left|\begin{array}{ccc}x& c& -b\\ -c& x& a\\ b& -a& x\end{array}\right|$
Cofactors of 1st Row of D are
$x^2+a^2,ab+cx,ac-bx$
Cofactor of 2nd Row of D are
$ab-cx,x^2+b^2,ax+bc$
and cofactors of 3rd row of D are
$ax+bx,bc-ax,x^2+c^2$
$\therefore$ Determinant of cofactors of D is
$D^c=\left|\begin{array}{ccc}{x}^2+a^2& ab+cx& ac-bx\\ ab-cx& x^2+b^2& ax+bc\\ ac+bx& bc-ax& x^2+c^2\end{array}\right|$
$=\left|\begin{array}{ccc}{a}^2+x^2& ab-cx& ac+bx\\ ab+cx& b^2+x^2& bc-ax\\ ac-bx& ax+bc& c^2+x^2\end{array}\right|$ (Rows interchanging into columns)
$=D^2$
$={\left|\begin{array}{ccc}x& c& -b^2\\ -c& x& a\\ b& -a& x\end{array}\right|}^2$
($\because D^c=D^2$, D is third order determinant)
Hence,
$\left|\begin{array}{ccc}{a}^2+x^2& ab-cx& ac+bx\\ ab+cx& b^2+x^2& bc-ax\\ ac-bx& ax+bc& c^2+x^2\end{array}\right|={\left|\begin{array}{ccc}x& c& -b\\ -c& x& a\\ b& -a& x\end{array}\right|}^2$