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 ___.
A
1
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B
2
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C
3
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D
4
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Solution
The correct option is D 4 Given, A=⎡⎢⎣abcbcacab⎤⎥⎦,abc=1andATA=INow,ATA=1⇒⎡⎢⎣abcbcacab⎤⎥⎦⎡⎢⎣abcbcacab⎤⎥⎦=⎡⎢⎣100010001⎤⎥⎦⎡⎢⎣a2+b2+c2ab+bc+caab+bc+caab+bc+caa2+b2+c2ab+bc+caab+bc+caab+bc+caa2+b2+c2⎤⎥⎦=⎡⎢⎣100010001⎤⎥⎦⇒a2+b2+c2=1andab+bc+ca=0.....(ii)weknowa3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca)⇒a3+b3+c3=(a+b+c)(1−0)+3[FromEqs.(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+3⇒a3+b3+c3=4.