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Using properties of determinants prove that:
\[
\left|\begin{array}{ccc}
1+a^{2}-b^{2} & 2 a b & -2 b \\
2 a b & 1-a^{2}+b^{2} & 2 a \\
2 b & -2 a & 1-a^{2}-b^{2}
\end{array}\right|=\left(1+a^{2}+b^{2}\right)^{3} .
\]
\( \mathrm{P} \)
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