Question

1 w w27 lw w? 17 If A= w w? 1 and B=w2 1 w , where w is a complex cube root of unity, then Tw? 1 w 1 w w?] prove that AB and BA both are null matrices.

JEE/Engineering Exams

Maths

Solution

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( A=left[begin{array}{lll}1 & 0 & omega w & w^{2} & 1 w^{2} & 1 & wend{array}right] quad B=left[begin{array}{lll}w & w^{2} & 1 w^{2} & 1 & w 1 & w & w^{2}end{array}right] ) ( [1] Rightarrow 4+left[C_{2}+C_{3}right] ) We know ( left[1+omega+omega^{2}=0right] ) (for Both A and B) ( A=left[begin{array}{ccc}1+w+omega^{2} & w+w^{2}+1 & w^{2}+1+w w & w^{2} & 1 w^{2} & 1 & wend{array}right] quad B=left[begin{array}{ccc}w+w^{2}+1 & w^{2}+1+60 & 1+w+w^{2} w^{2} & 1 & w 1 & omega & w^{2}end{array}right] ) ( A=left[begin{array}{lll}0 & 0 & 0 omega & omega^{2} & 1 omega^{2} & 1 & omegaend{array}right] quad B=left[begin{array}{ccc}0 & 0 & 0 omega^{2} & 1 & omega 1 & omega & omega^{2}end{array}right] ) ( therefore quad A=0 quad B=0 ) Both are null makicy Herra Proved