Question
Let $$a, b, c$$ be such that $$b\left( {a + c} \right) \ne 0$$ if \[\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,\,a - 1\\
- b\,\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,b - 1\\
\,\,c\,\,\,\,\,\,\,\,\,c - 1\,\,\,\,\,\,\,\,\,\,c + 1
\end{array} \right| + \left| \begin{array}{l}
\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c - 1\\
\,\,\,\,\,a - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c + 1\\
{\left( { - 1} \right)^{n + 2}}a\,\,\,\,\,\,\,\,{\left( { - 1} \right)^{n + 1}}b\,\,\,\,\,\,\,\,\,{\left( { - 1} \right)^n}c
\end{array} \right| = 0,\] then the value of $$n$$ is:
A.
any even integer
B.
any odd integer
C.
any integer
D.
zero
Answer :
any odd integer
Solution :
\[\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,\,a - 1\\
- b\,\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,b - 1\\
\,\,c\,\,\,\,\,\,\,\,\,c - 1\,\,\,\,\,\,\,\,\,\,c + 1
\end{array} \right| + \left| \begin{array}{l}
\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c - 1\\
\,\,\,\,\,a - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c + 1\\
{\left( { - 1} \right)^{n + 2}}a\,\,\,\,\,\,\,\,{\left( { - 1} \right)^{n + 1}}b\,\,\,\,\,\,\,\,\,{\left( { - 1} \right)^n}c
\end{array} \right| = 0\]
\[ \Rightarrow \,\,\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,\,a - 1\\
- b\,\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,b - 1\\
\,\,c\,\,\,\,\,\,\,\,\,c - 1\,\,\,\,\,\,\,\,\,\,c + 1
\end{array} \right| + \left| \begin{array}{l}
a + 1\,\,\,\,\,\,\,\,\,\,a - 1\,\,\,\,\,\,\,\,\,\,\,{\left( { - 1} \right)^{n + 2}}a\\
b + 1\,\,\,\,\,\,\,\,\,b - 1\,\,\,\,\,\,\,\,\,\,\,\,{\left( { - 1} \right)^{n + 1}}b\\
c - 1\,\,\,\,\,\,\,\,\,c + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\left( { - 1} \right)^n}c
\end{array} \right| = 0\]
(Taking transpose of second determinant)
$${C_1} \Leftrightarrow {C_3}$$
\[ \Rightarrow \,\,\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,a - 1\\
- b\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,b - 1\\
\,\,c\,\,\,\,\,\,\,\,c - 1\,\,\,\,\,\,\,\,\,\,c + 1
\end{array} \right| - \left| \begin{array}{l}
\,\,\,{\left( { - 1} \right)^{n + 2}}a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,a + 1\\
{\left( { - 1} \right)^{n + 2}}\left( { - b} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,b - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,b + 1\\
\,\,\,{\left( { - 1} \right)^{n + 2}}c\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,c - 1
\end{array} \right| = 0\]
$${C_2} \Leftrightarrow {C_3}$$
\[ \Rightarrow \,\,\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,a - 1\\
- b\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,b - 1\\
\,\,c\,\,\,\,\,\,\,\,c - 1\,\,\,\,\,\,\,\,\,\,c + 1
\end{array} \right| + {\left( { - 1} \right)^{n + 2}}\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,a - 1\\
- b\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,b - 1\\
\,\,c\,\,\,\,\,\,\,\,c - 1\,\,\,\,\,\,\,\,\,\,c + 1
\end{array} \right| = 0\]
\[ \Rightarrow \,\,\left[ {1 + {{\left( { - 1} \right)}^{n + 2}}} \right]\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,a + 1\,\,\,\,\,\,\,\,a - 1\\
- b\,\,\,\,\,\,\,b + 1\,\,\,\,\,\,\,\,\,b - 1\\
\,\,c\,\,\,\,\,\,\,\,c - 1\,\,\,\,\,\,\,\,\,\,c + 1
\end{array} \right| = 0\]
$${C_2} - {C_1},{C_3} - {C_1}$$
\[ \Rightarrow \,\,\left[ {1 + {{\left( { - 1} \right)}^{n + 2}}} \right]\left| \begin{array}{l}
\,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 1\\
- b\,\,\,\,\,\,\,2b + 1\,\,\,\,\,\,\,\,\,2b - 1\\
\,\,c\,\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1
\end{array} \right| = 0\]
$${R_1} + {R_3}$$
\[ \Rightarrow \,\,\left[ {1 + {{\left( { - 1} \right)}^{n + 2}}} \right]\left| \begin{array}{l}
a + c\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\
\, - b\,\,\,\,\,\,\,\,\,2b + 1\,\,\,\,\,\,\,\,\,2b - 1\\
\,\,\,\,c\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1
\end{array} \right| = 0\]
$$\eqalign{
& \Rightarrow \,\,\left[ {1 + {{\left( { - 1} \right)}^{n + 2}}} \right]\left( {a + c} \right)\left( {2b + 1 + 2b - 1} \right) = 0 \cr
& \Rightarrow \,\,4b\left( {a + c} \right)\left[ {1 + {{\left( { - 1} \right)}^{n + 2}}} \right] = 0 \cr
& \Rightarrow \,\,1 + {\left( { - 1} \right)^{n + 2}} = 0\,\,{\text{as }}b\left( {a + c} \right) \ne 0 \cr} $$
⇒ $$n$$ should be an odd integer.