Question

The number of $3\times 3$ non-singular matrices with four entries as $1$ and all other entries as $0$ is

• A. 6
• B. at least 7
• C. less than 4
• D. 5

Answer 1

Answer: (B)

The matrix $\left(\begin{array}{ccc}1& a& b\\ c& 1& d\\ e& f& 1\end{array}\right)$ where exectly one of $a,b,c,d,e,f$ is 1 and rest of them are zero, is invertible. There are six such matrices.
Also the matrix $\left|\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right|$ is invertible Thus, there are at least 7 such matrices which are invertible.