Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

The matrices in the form $$\left[\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right],a_{ij}\in \{0,1,2\},a_{11}=a_{12}\text{ are }$$ $$\left[\begin{array}{cc}0& 0/1/2\\ 0/1/2& 0\end{array}\right],\left[\begin{array}{cc}1& 0/1/2\\ 0/1/2& 1\end{array}\right],\left[\begin{array}{cc}2& 0/1/2\\ 0/1/2& 2\end{array}\right]$$ 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 $=27-7=20$