Question

Let $S = \left\{\begin{pmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{pmatrix} : 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
$\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix} , a_{ij} \in \left\{0,1,2\right\} , a_{11} = a_{12}$ are $ \begin{bmatrix}0&0/1/2\\ 0/1/2&0\end{bmatrix}, \begin{bmatrix}1&0/1/2\\ 0/1/2&1\end{bmatrix}, \begin{bmatrix}2&0/1/2\\ 0/1/2&2\end{bmatrix} $
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
$\begin{bmatrix}0&0/1/2\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 1/2&0\end{bmatrix},\begin{bmatrix}1&1\\ 1&1\end{bmatrix}, \begin{bmatrix}2&2\\ 2&2\end{bmatrix}$
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