Question

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

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

Answer 1

The correct option is D

at least $7$

Find the number of matrices formed based on given conditions

Form the matrix,

$\left[\begin{array}{ccc}1& *& *\\ *& 1& *\\ *& *& 1\end{array}\right]$

We have four $1$'s. In the above matrix, three are filled. So, remaining $1$ can be placed in any of the $6$$*$ positions and $0$ elsewhere.

Therefore, we get $6$ non-singular matrices

Now consider

$\left[\begin{array}{ccc}*& *& 1\\ *& 1& *\\ 1& *& *\end{array}\right]$

Similarly we get another set of $6$ non-singular matrices.

So, we can easily form multiple matrices based on given conditions.

Hence, the correct option is $\left(\mathrm{D}\right)$.