Question

Let $a, b, c$ be such that $b(a + c)$ $\neq$ 0 if $\begin{vmatrix}a&a+1&a-1\\ -b &b+1&b-1\\ c &c-1&c+1\end{vmatrix} + \begin{vmatrix}a+1&b+1&c-1\\ a-1 &b-1&c+1\\ \left(-1\right)^{n+2}a &\left(-1\right)^{n+1}b&\left(-1\right)^{n}c\end{vmatrix}= 0 $ then the value of $n$ is :

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

Answer 1

Answer: (B)

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