Let S= ( array ll a₁₁ & a₁₂ a₂₁ & a₂₂ array ): a_ i j 0,1,2 , a₁₁=a₂₂ Then the number of non-singular matrices in the set S is :

Step-by-step solution

The matrices in the form [ array ll a₁₁ & a₁₂ a₂₁ & a₂₂ array ], a_ i j 0,1,2 , a₁₁=a₁₂ are [ array cc 0 & 0 / 1 / 2 0 / 1 / 2 & 0 array ], [ array cc 1 & 0 / 1 / 2 0 / 1 / 2 & 1 array ], [ array cc 2 & 0 / 1 / 2 0 / 1 / 2 & 2 array ] At any place, 0 / 1 / 2 means 0,1 or 2 will be the element at that place. Hence there are total 27=3 3+3 3+3 3 ) matrices of the above form. Out of which the matrices which are singular are [ array cc 0 & 0 / 1 / 2 0 & 0 array ], [ array cc 0 & 0 1 / 2 & 0 array ], [ array ll 1 & 1 1