Let S=\ ( array ll a_ 11 & a_ 12 a_ 21 & a_ 22 array ): a_ i j ∈\...

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 } \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] At any place, $0 / 1 / 2$ means 0,1 or 2 will be the element at that place. Hence there are total $27=3 \times 3+3 \times 3+3 \times 3$ ) 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] Hence there are total $7(=3+2+1+1)$ singular matrices. Therefore number of all non-singular matrices in the given form $=27-7=20$