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Let $a ,b ,c $ be such that $b + c \ne 0$ if
$\left| {\begin{array}{*{20}{c}}a&{a + 1}&{a - 1}\\{ - b}&{b + 1}&{b - 1}\\c&{c - 1}&{c + 1}\end{array}} \right| + \left| {\begin{array}{*{20}{c}}{a + 1}&{b + 1}&{c - 1}\\{a - 1}&{b - 1}&{c + 1}\\{{{\left( { - 1} \right)}^{n + 2}} \cdot a}&{{{\left( { - 1} \right)}^{n + 1}} \cdot b}&{{{\left( { - 1} \right)}^n} \cdot c}\end{array}} \right| = 0$ then $n$ equals to
Zero
any even integer
any odd integer
any integer
Solution
$\left| {\begin{array}{*{20}{c}}
a&{a + 1}&{a – 1}\\
{ – b}&{b + 1}&{b – 1}\\
c&{c – 1}&{c + 1}
\end{array}} \right| + \left| {\begin{array}{*{20}{c}}
{{{\left( { – 1} \right)}^{n + 2}}a}&{a + 1}&{a – 1}\\
{{{\left( { – 1} \right)}^{n + 1}}b}&{b + 1}&{b – 1}\\
{{{\left( { – 1} \right)}^n}c}&{c – 1}&{c + 1}
\end{array}} \right|$
$ = \left| {\begin{array}{*{20}{c}}
{a + {{\left( { – 1} \right)}^{n + 2}}a}&{a + 1}&{a – 1}\\
{ – b + {{\left( { – 1} \right)}^{n + 1}}b}&{b + 1}&{b – 1}\\
{c + {{\left( { – 1} \right)}^n}c}&{c – 1}&{c + 1}
\end{array}} \right|$
$=0$ if $n$ is an obb integer.
