If ω is cube root of unity and $$x+y+z=a$$, $$x+$$ω$$y+$$ω$$2z=b$$, $$x+$$ω$$2y+$$ω$$z=c$$ then which of the following is correct

Question

If ω is cube root of unity and $$x+y+z=a$$,   $$x+$$ω$$y+$$ω$$2z=b$$, $$x+$$ω$$2y+$$ω$$z=c$$ then which of the following is correct

Infinity Learn · Accepted Answer

The determinant of the coefficient matrix is Δ$$=1111$$ωω$$21$$ω$$2$$ω$$=1$$ω$$2-$$ω$$4-1$$ω$$-$$ω$$2+1$$ω$$2-$$ω$$=3$$ω$$2-$$ω ∵ω$$4=$$ω Also Δ$$1=a11b$$ωω$$2c$$ω$$2$$ω$$=a$$ω$$2-$$ω$$4-b$$ω$$-$$ω$$2+c$$ω$$2-$$ω$$=a+b+c$$ω$$2-$$ω            Δ$$2=1a11b$$ω$$21c$$ω$$=1b$$ω$$-c$$ω$$2-1a$$ω$$-c+1a$$ω$$2-cb=a+b$$ω$$2+c$$ωω$$2-$$ω    and Δ$$3=11a1$$ω$$b1$$ω$$2c=1c$$ω$$-b$$ω$$2-1c-a$$ω$$2+1b-a$$ω$$=a+b$$ω$$+c$$ω$$2$$ω$$2-$$ω ∴By Crammar's rule we have, $$x=$$Δ$$1$$Δ,$$y=$$Δ$$2$$Δand $$z=$$Δ$$3$$Δ. ⇒$$x=a+b+c3$$, $$y=a+b$$ω$$2+c$$ω$$3$$ and $$z=a+b$$ω$$+c$$ω$$23$$

If $ω$ is cube root of unity and $x + y + z = a, x + ω y + ω^{2} z = b$ , $x + ω^{2} y + ω z = c$ then which of the following is correct

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If ω is cube root of unity and x+y+z=a, x+ωy+ω2z=b, x+ω2y+ωz=c then which of the following is correct

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If $ω$ is cube root of unity and $x + y + z = a, x + ω y + ω^{2} z = b$ , $x + ω^{2} y + ω z = c$ then which of the following is correct