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If $f(x)=\left|\begin{array}{ccc}x^{3}-x & a+x & b+x \\ x-a & x^{2}-x & c+x \\ x-b & x-c & 0\end{array}\right|$, then
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Verified Answer
The correct answer is:
$f(0)=0$
We have,
$\begin{aligned}
f(x) &=\left|\begin{array}{ccc}
x^{3}-x & a+x & b+x \\
x-a & x^{2}-x & c+x \\
x-b & x-c & 0
\end{array}\right| \\
f(0) &=\left|\begin{array}{ccc}
0 & a & b \\
-a & 0 & c \\
-b & -c & 0
\end{array}\right| \\
f(0)=0 &
\end{aligned}$
$f(0)$ is skew symmetric matrix of order $3 .$
$\begin{aligned}
f(x) &=\left|\begin{array}{ccc}
x^{3}-x & a+x & b+x \\
x-a & x^{2}-x & c+x \\
x-b & x-c & 0
\end{array}\right| \\
f(0) &=\left|\begin{array}{ccc}
0 & a & b \\
-a & 0 & c \\
-b & -c & 0
\end{array}\right| \\
f(0)=0 &
\end{aligned}$
$f(0)$ is skew symmetric matrix of order $3 .$
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