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Question

If matrix A=abcbcacab where a, b, c are real positive numbers, abc = 1 and ATA=I, then the value of a3+b3+c3 is ___

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Solution

Given, A=abcbcacab,abc=1 and AT A=INow,AT A=1abcbcacababcbcacab=100010001a2+b2+c2ab+bc+caab+bc+caab+bc+caa2+b2+c2ab+bc+caab+bc+caab+bc+caa2+b2+c2=100010001a2+b2+c2=1 and ab+bc+ca=0.....(ii)we know a3+b3+c33abc=(a+b+c)(a2+b2+c2abbcca)a3+b3+c3=(a+b+c)(10)+3[From Eqs.(i)and(iii)]a3+b3+c3=(a+b+c)+3.......(iii)Now,(a+b+c)2=a2+b2+c2+2(ab+bc+ca)=1......(iv)From,Eq.(iii),a3+b3+c3=1+3a3+b3+c3=4.


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