Question

If $a,b,c$ are real numbers such that $a-b=1,b-c=3$, then the number of matrices of the form $A=\left[\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right]$ such that $|A|=-12$, is

• A. 1
• B. 2
• C. 3
• D. infinitely many

Answer 1

Answer: (D)

We have,
$A=\left[\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right]$
$|A|=\left[\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right]$
$|A|=\left[\begin{array}{ccc}1& 0& 0\\ a& b-a& c-a\\ a^2& b^2-a^2& c^2-a^2\end{array}\right]=(b-a)(c-a)(c-b)$
$|A|=(a-b)(b-c)(c-a)$
$|A|=-12,(a-b)=1,(b-c)=3$
$\because (a-b)(b-c)(c-a)=-12$
$1\times 3\times (c-a)=-12$
$c-a=-4$
$\Rightarrow a-c=4$
$\because a-b=1,b-c=3,a-c=4$
$\therefore$ Infinite solution.