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Question: If u = ax + by + cz, v = ay + bz + cx, w = ax + bx + cy then the value of \(\left| \begin{matrix} a ...

If u = ax + by + cz, v = ay + bz + cx, w = ax + bx + cy then the value of abcbcacab\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right| × xyzyzxzxy\left| \begin{matrix} x & y & z \\ y & z & x \\ z & x & y \end{matrix} \right| is –

A

u2 + v2 + w2 – 2uvw

B

u3 + v3 + w3 – 3uvw

C

0

D

None of these

Answer

u3 + v3 + w3 – 3uvw

Explanation

Solution

D1D2= ax+by+czay+bz+cxaz+bx+cybx+cy+azby+cz+axbz+cx+aycx+ay+bzcy+az+bxcz+ax+by\left| \begin{matrix} ax + by + cz & ay + bz + cx & az + bx + cy \\ bx + cy + az & by + cz + ax & bz + cx + ay \\ cx + ay + bz & cy + az + bx & cz + ax + by \end{matrix} \right|

= uvwwuvvwu\left| \begin{matrix} u & v & w \\ w & u & v \\ v & w & u \end{matrix} \right|

= u(u2 – vw) – v(wu – v2) + w(w2 – uv)

= u3 + v3