Let S= ( … ): a_i j ∈0,1,2, a_11=a_22 Then the number of non-singular matrices in the set S is :

The matrices in the form $$\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], a_{i j} \in\{0,1,2\}, a_{11}=a_{12} \text { are }$$[a11​a21​​a12​a22​​],aij​∈{0,1,2},a11​=a12​ are  $$\left[\begin{array}{cc} 0 & 0 / 1 / 2 \\ 0 / 1 / 2 & 0 \end{array}\right],\left[\begin{array}{cc} 1 & 0 / 1 / 2 \\ 0 / 1 / 2 & 1 \end{array}\right],\left[\begin{array}{cc} 2 & 0 / 1 / 2 \\ 0 / 1 / 2 & 2 \end{array}\right]$$[00/1/2​0/1/20​],[10/1/2​0/1/21​],[20/1/2​0/1/22​] At any place, $0 / 1 / 2$0/1/2 means 0,1 or 2 will be the element at that place. Hence there are total $27=3 \times 3+3 \times 3+3 \times 3$27=3×3+3×3+3×3 ) matrices of the above form. Out of which the matrices which are singular are $$\left[\begin{array}{cc} 0 & 0 / 1 / 2 \\ 0 & 0 \end{array}\right],\left[\begin{array}{cc} 0 & 0 \\ 1 / 2 & 0 \end{array}\right],\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right],\left[\begin{array}{ll} 2 & 2 \\ 2 & 2 \end{array}\right]$$[00​0/1/20​],[01/2​00​],[11​11​],[22​22​] Hence there are total $7(=3+2+1+1)$7(=3+2+1+1) singular matrices. Therefore number of all non-singular matrices in the given form $=27-7=20$=27−7=20