यदि p + q + r = 0 हो, तो सिद्ध कीजिए कि |(pa qb rc),(qc ra pb),(rb pc qa)| = pqr|(a b c),(c a b),(b c a)|

Question

यदि p + q + r = 0 हो, तो सिद्ध कीजिए कि

  \(\begin{vmatrix}pa&qb&rc\\ qc&ra&pb\\ rb&pc&qa\end{vmatrix}\) = pqr    \(\begin{vmatrix}a&b&c\\c&a&b\\b&c&a\end{vmatrix}\)

Answer 1

L.H.S. = \(\begin{vmatrix}pa&qb&rc\\ qc&ra&pb\\ rb&pc&qa\end{vmatrix}\) 

= pa(qra² – p²bc) – qb(q²ca – prb²) + rc(pqc²– r²ab)

= pqra³ – p³abc – q³abc + pqrb³ + pqrc³ – r³abc

= a³pqr – p³abc – q³abc + b³pqr + c³pqr – r³abc

= pqr(a³ + b³ + c³) – abc(p³ + q³ + r³)

= pqr(a³ + b³ + c³) – abc(3pqr)

(∵ p + q+ r = 0 ⇒ p³ + q³ + r³ = 3pqr)

= pqr[a³ + b³ + c³ – 3abc] …(i)

R.H.S. =pqr  \(\begin{vmatrix}a&b&c\\c&a&b\\b&c&a\end{vmatrix}\)

= pqr[a(a²– bc) – b(ca – b²) + c(c²– ab)]

= pqr[a – abc – abc + b³ + c³ – abc]

= pqr[a³ + b³ + c³ – 3abc]

समीकरण (i) व (i) से,

 pqr = \(\begin{vmatrix}a&b&c\\c&a&b\\b&c&a\end{vmatrix}\)

इति सिद्धम्।