A) 27
B) 24
C) 10
D) 20
Correct Answer: D
Solution :
The matrices in the form. \[\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} \\ {{a}_{21}} & {{a}_{22}} \\ \end{matrix} \right],{{a}_{ij}}\in \{0,1,2\},{{a}_{11}}={{a}_{12}}\]are \[\left[ \begin{matrix} 0 & 0/1/2 \\ 0/1/2 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 0/1/2 \\ 0/1/2 & 1 \\ \end{matrix} \right],\left[ \begin{matrix} 2 & 0/1/2 \\ 0/1/2 & 2 \\ \end{matrix} \right]\]At anyplace, 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 \[\left[ \begin{matrix} 0 & 0/1/2 \\ 0 & 0 \\ \end{matrix} \right],\left[ \begin{matrix} 0 & 0 \\ 1/2 & 0 \\ \end{matrix} \right],\left[ \begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right],\left[ \begin{matrix} 2 & 2 \\ 2 & 2 \\ \end{matrix} \right]\] 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 .
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