# The number of 3x3 non singular matrices, with four entries as 1 and all other entries are 0 ,is? a)5 b)6 c)at least 7 d) less than 4

Dec 19, 2017

There are exactly $36$ such non-singular matrices, so c) is the correct answer.

#### Explanation:

First consider the number of non-singular matrices with $3$ entries being $1$ and the rest $0$.

They must have one $1$ in each of the rows and columns, so the only possibilities are:

$\left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right) \text{ "((1, 0, 0), (0, 0, 1), (0, 1, 0))" } \left(\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{matrix}\right)$

$\left(\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{matrix}\right) \text{ "((0, 0, 1), (1, 0, 0), (0, 1, 0))" } \left(\begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{matrix}\right)$

For each of these $6$ possibilities we can make any one of the remaining six $0$'s into a $1$. These are all distinguishable. So there are a total of $6 \times 6 = 36$ non-singular $3 \times 3$ matrices with $4$ entries being $1$ and the remaining $5$ entries $0$.