Let $a,b,c$ be such that $(b + c)
eq 0$ and $\left| \begin{matrix} a & a + 1 & a - 1 \ - b & b + 1 & b - 1 \ c & c - 1 & c + 1 \end{matrix} ight|$ $+ \left| \begin{matrix} a + 1 & b + 1 & c - 1 \ a - 1 & b - 1 & c + 1 \ ( - 1)^{n + 2}a & ( - 1)^{n + 1}b & ( - 1)^{n}c \end{matrix} ight|^{1} = 0$ Then the value of $n$ is

Question

Let $a,b,c$ be such that $(b + c) 
eq 0$ and $\left| \begin{matrix} a & a + 1 & a - 1 \ - b & b + 1 & b - 1 \ c & c - 1 & c + 1 \end{matrix} ight|$ $+ \left| \begin{matrix} a + 1 & b + 1 & c - 1 \ a - 1 & b - 1 & c + 1 \ ( - 1)^{n + 2}a & ( - 1)^{n + 1}b & ( - 1)^{n}c \end{matrix} ight|^{1} = 0$ Then the value of $n$ is

Dr. Abubacker · Accepted Answer

(c)
$\left| \begin{matrix}
a & a + 1 & a - 1 \
 - b & b + 1 & b - 1 \
c & c - 1 & c + 1
\end{matrix} ight| + ( - 1)^{n}\left| \begin{matrix}
a + 1 & b + 1 & c - 1 \
a - 1 & b - 1 & c + 1 \
a & - b & c
\end{matrix} ight|^{1}$
$= \left| \begin{matrix}
a & a + 1 & a - 1 \
 - b & b + 1 & b - 1 \
c & c - 1 & c + 1
\end{matrix} ight| + ( - 1)^{n}\left| \begin{matrix}
a + 1 & a - 1 & a \
b + 1 & b - 1 & - b \
c - 1 & c + 1 & c
\end{matrix} ight|$
$= = \left| \begin{matrix}
a & a + 1 & a - 1 \
 - b & b + 1 & b - 1 \
c & c - 1 & c + 1
\end{matrix} ight| + ( - 1)^{n + 1}\left| \begin{matrix}
a + 1 & a & a - 1 \
b + 1 & - b & b - 1 \
c - 1 & c & c + 1
\end{matrix} ight|$
$C_{2} \leftrightarrow C_{3}$
$= (1 + ( - 1)^{n + 2})\left| \begin{matrix}
a & a + 1 & a - 1 \
 - b & b + 1 & b - 1 \
c & c - 1 & c + 1
\end{matrix} ight|$
This is equal to zero only, if $n + 2$ is odd $ie,\ n$ is an odd integer.

Explanation

Reviewed by Dr. Abubacker

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