Proof:
i) (2f) (2g) (2h) | = | (m1n2 + m2n1) (l1n2 + l2n1) (l1m2 + l2m1) |
= | (m1n2 + m2n1) (l12m2n2 + 11l2m1n2 + 11l2m2n1 + l22m1 n1) | |
= | l12m1m2n22 + 11l2m12n22 + 11l2m1m2n1n2 + 122m12n1n2 + 112m22n1n2 + 11l2m1m2n1n2 + 11l2m22n12 + l22m1m2n12 | |
= | l1l2 (m12n22 + m22n12) + m1m2 (l12n22 + l22n12) + n1n2 (l12m22 + l22m12) + 2 l1l2 m1m2 n1n2 | |
= | l1l2 [(m1n2 + m2n1)2 – 2 m1n2m2n1 ] + m1m2 [(l1n2 + l2n1)2 – 2 l1n2l2n1 ] + n1n2 [(l1m2 + l2m1)2 – 2 l1m2l2m1 ] + 2 l1l2 m1m2 n1n2 | |
= | a [(2f)2 – 2bc] + b [(2g)2 – 2ac] + c [(2h)2 – 2ab] + 2abc | |
= | a [4f 2 – 2bc] + b [4g2 – 2ac] + c [4h2 – 2ab] + 2abc | |
= | 4af 2 – 2abc + 4bg2 – 2abc + 4ch2 – 2abc + 2abc | |
= | – 4abc + 4af 2 + 4bg2 + 4ch2 | |
⇒ 8fgh | = | – 4abc + 4af 2 + 4bg2 + 4ch2 |
ii) 4(h2 – ab) | = | (2h)2 – 4ab |
= | (l1m2 + l2m1)2 – 4 l1l2m1m2 | |
= | (l1m2 – l2m1)2 ≥ 0 |