G= [ array ll x & x x & x array ], x . is a non-zero real number is a group with respect to matrix multiplication. In this group, the inverse of [ array ll 1 3 & 1 3 1 3 & 1 3 array ] is

Step-by-step solution

Given, G= [ array ll x & x x & x array ] is a group with respect to matrix multiplication where x R- 0 . Now, the identity element of above group with respect to matrix x . Multiplication is = [ array ll 1 / 2 & 1 / 2 1 / 2 & 1 / 2 array ]=I^ For inverse; A A⁻¹=I^ Given, [ array ll 1 / 3 & 1 / 3 1 / 3 & 1 / 3 array ] A⁻¹= [ array ll 1 / 2 & 1 / 2 1 / 2 & 1 / 2 array ] Apply R₁ 3 / 2 R₁ and R₂ 3 / 2 R₂ gathered [ array ll 1 / 2 & 1 / 2 1 / 2 & 1 / 2 array ] A⁻¹= [ array ll 3 / 4 & 3 / 4 3 / 4 & 3 / 4 array ] I^ A¹=