The maximum value of the determinant of the matrix ( [ array ccc 1+ ^2 x & ^2 x & 4 2 x ^2 x & 1+ ^2 x & 4 2 x ^2 x & ^2 x & 1+4 2 x array ] is )

Step-by-step solution

Given, ( | array ccc 1+ ^2 x & ^2 x & 4 2 x ^2 x & 1+ ^2 x & 4 2 x ^2 x & ^2 x & 1+4 2 x array | ) Applying (R₁ R₁-R₃ ) and (R₂ R₂-R₃ ), we get (= | array ccc 1 & 0 & -1 0 & 1 & -1 ^2 x & ^2 x & 1+4 2 x array | ) Applying (C₂ C₂+C₁ ) (= | array ccc 1 & 1 & -1 0 & 1 & -1 ^2 x & 1 & 1+4 2 x array | ) Applying (R₂ R₂-R₁ ) and (R₃ R₃-R₁ ) (= | array ccc 1 & 1 & -1 -1 & 0 & 0 -1+ ^2 x & 0 & 2+4 2 x array |=2+4 2 x ) Since, maximum value of ( 2 x ) is 1. (=2+4=6 )