Question

Let $S=\left\{\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right):a_{ij}\in \left\{0,1,2\right\},a_{11}=a_{22}\right\}$
Then the number of non-singular matrices in the set S is :

• A. 27
• B. 24
• C. 10
• D. 20

Answer 1

Answer: (D)

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