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Standard XII
Mathematics
Properties of Determinants
Prove | [ a...
Question

Prove ∣∣ ∣ ∣∣(a−p)2(a−q)2(a−r)2(b−p)2(b−q)2(b−r)2(c−p)2(c−q)2(c−r)2∣∣ ∣ ∣∣=∣∣ ∣ ∣∣(1+ap)2(1+aq)2(1+ar)2(1+bq)2(1+bq)2(1+br)2(1+cp)2(1+cq)2(1+cr)2∣∣ ∣ ∣∣

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Solution

L.H.S=∣∣ ∣ ∣∣(a−p)2(a−q)2(a−r)2(b−p)2(b−q)2(b−r)2(c−p)2(c−q)2(c−r)2∣∣ ∣ ∣∣=∣∣ ∣ ∣∣a2−2ap+p2b2−2bp+p2c2−2cp+p2a2−2aq+q2b2−2bq+q2c2−2cq+q2a2−2aq+q2b2−2bq+q2c2−2cr+r2∣∣ ∣ ∣∣=∣∣ ∣ ∣∣1pp21qq21rr2∣∣ ∣ ∣∣×∣∣ ∣ ∣∣12aa212bb212cc2∣∣ ∣ ∣∣=∣∣ ∣ ∣∣1+2ap+a2p21+2bp+b2p21+2cp+c2p21+2aq+a2q21+2bq+b2q21+2cq+c2q21+2ar+a2r21+2br+b2r21+2cr+c2r2∣∣ ∣ ∣∣=∣∣ ∣ ∣∣(1+ap)2(1+bp)2(1+cp)2(1+aq)2(1+bq)2(1+cq)2(1+ar)2(1+br)2(1+cr)2∣∣ ∣ ∣∣=∣∣ ∣ ∣∣(1+ap)2(1+aq)2(1+ar)2(1+bq)2(1+bq)2(1+br)2(1+cp)2(1+cq)2(1+cr)2∣∣ ∣ ∣∣=R.H.SHence Proved

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