Suppose $\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}$ exists and is equal to L, where

$$F(x) = \left| {\matrix{ {a + \sin {x \over 2}} & { - b\cos x} & 0 \cr { - b\cos x} & 0 & {a + \sin {x \over 2}} \cr 0 & {a + \sin {x \over 2}} & { - b\cos x} \cr } } \right|$$.

Then, $-$112 L is equal to ___________.