If is an imaginary cube root of unity, then the value of [ array ccc 1 & ² & 1- ⁴ & 1 & 1+ ⁵ 1 & & ² array ] is

Step-by-step solution

Given, ³=1 and 1+ + ²=0 = | array ccc 1 & ² & 1- ⁴ & 1 & 1+ ⁵ 1 & & ² array |= | array ccc 1 & ² & 1- & 1 & 1+ ² 1 & & ² array | Expand with respect to R₁ aligned &= ( ²- - ³ )- ² ( ³-1- ² ) & +(1- ) ( ²-1 ) &= ( ²- -1 )- ² (1-1- ² ) & +(1- ) ( ²-1 ) &= ²- -1+ ⁴+ ²-1- ³+ &=2 ²-2+ -1=2 ²+ -3 &= ²+ ( + ² )-3 &= ²-1-3= ²-4 aligned