Manipal MET2014MathematicsComplex Number
If a, b, c are cube roots of unity, then | array lll e^a & e^ 2 a & e^ 3 a -1 e^b & e^ 2 b & e^ 3 b -1 e^c & e^ 2 c & e^ 3 c -1 array | is equal to
Options
- A0
- Be
- Ce^2
- De^3
Correct answer
A. 0
Step-by-step solution
aligned & = | array lll e^a & e^ 2 a & e^ 3 a e^b & e^ 2 b & e^ 3 b e^c & e^ 2 c & e^ 3 c array |- | array lll e^a & e^ 2 a & 1 e^b & e^ 2 b & 1 e^c & e^ 2 c & 1 array | & & =e^a e^b e^c | array lll 1 & e^a & e^ 2 a 1 & e^b & e^ 2 b 1 & e^c & e^ 2 c array |+ | array lll e^a & 1 & e^ 2 a e^b & 1 & e^ 2 b e^c & 1 & e^ 2 c array | & & =e^ (a+b+c) | array lll 1 & e^a & e^ 2 a 1 & e^b & e^ 2 b 1 & e^c & e^ 2 c array |- | array lll 1 & e^a & e^ 2 a 1 & e^b & e^ 2 b 1 & e^c & e^ 2 c array | & & = (e^ a+b+c -1 ) | array ll