Question

Let a, b and c be such that b(a+c) $\ne$ 0.
If $\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ (-1{)}^{n+2}a& (-1{)}^{n-1}b& (-1{)}^{n}c\end{array}\right|=0$, then the value of n is

• A. zero
• B. any even integer
• C. any odd integer
• D. any integer

Answer 1

The correct option is C any odd integer

$\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+(-1{)}^n\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ a& -b& c\end{array}\right|$
$=\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+(-1{)}^n\left|\begin{array}{ccc}a+1& a-1& a\\ b+1& b-1& -b\\ c-1& c+1& c\end{array}\right|$
$=\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+(-1{)}^{n+2}\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|$
This is equal to zero only if n + 2 is odd, i.e, n is an odd integer.