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Show that the determinant
∣ ∣ ∣a2+b2+c2bc+ca+abbc+ca+abbc+ca+aba2+b2+c2bc+ca+abbc+ca+abbc+ca+aba2+b2+c2∣ ∣ ∣
is always non-negative.

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Solution

∣ ∣ ∣a2+b2+c2bc+ca+abbc+ca+abbc+ca+aba2+b2+c2bc+ca+abbc+ca+abbc+ca+aba2+b2+c2∣ ∣ ∣=∣ ∣abcbcacab∣ ∣∣ ∣abcbcacab∣ ∣=∣ ∣abcbcacab∣ ∣2=(a+b+c)2(a2+b2+c2abbcca)2>0
for all a,b,c,abc
it always non-negative.(positive).

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