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)

at least 7

Answer 2

The matrix $\begin{pmatrix}1&a&b\\\ c&1&d\\\ e&f&1\end{pmatrix}$ 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 $\begin{vmatrix}1&0&1\\\ 0&1&0\\\ 1&0&0\end{vmatrix}$ is invertible Thus, there are at least 7 such matrices which are invertible.