Without expanding, prove that `|a+b x c+dx p+q x a x+b c x+d p x+q u v w|=(1-x^2)|a c p b d q u v w|` .

Answer

Proved

Explanation

Let `|A|=|(a+b x, c+d x, p+q x),(a x+b, c x+d, p x+q),(u, v, w)|`


Apply `R_{2} \rightarrow R_{2}-x R_{1}`
$$
|A|=\left|\begin{array}{ccc}
a+b x & c+d x & p+q x \\
b-b x^{2} & d-d x^{2} & q-q x^{2} \\
u & v & w
\end{array}\right|
...