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Let $a, b, c$ be such that $b(a+c) \neq 0$. 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
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1072 Upvotes
Verified Answer
The correct answer is:
any odd integer
any odd integer
$$
\begin{aligned}
& \left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right|+(-1)^n\left|\begin{array}{ccc}
a+1 & b+1 & c-1 \\
a-1 & b-1 & c+1 \\
a & -b & c
\end{array}\right|=\left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right|+(-1)^n\left|\begin{array}{cccc}
a+1 & a-1 & a \\
b+1 & b-1 & -b \\
c-1 & c+1 & c
\end{array}\right| \\
& =\left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right|+(-1)^{n+1}\left|\begin{array}{ccc}
a+1 & a & a-1 \\
b+1 & -b & b-1 \\
c-1 & c & c+1
\end{array}\right|=\left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right|+(-1)^{n+2}\left|\begin{array}{ccc}
a & a+1 & a-1 \\
-b & b+1 & b-1 \\
c & c-1 & c+1
\end{array}\right|
\end{aligned}
$$
This is equal to zero only if $n+2$ is odd i.e. $n$ is odd integer.
\begin{aligned}
& \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}{cccc}
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+1}\left|\begin{array}{ccc}
a+1 & a & a-1 \\
b+1 & -b & b-1 \\
c-1 & c & c+1
\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|
\end{aligned}
$$
This is equal to zero only if $n+2$ is odd i.e. $n$ is odd integer.
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