Question

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

Answer 1

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:

`((1, 0, 0), (0, 1, 0), (0, 0, 1))" "((1, 0, 0), (0, 0, 1), (0, 1, 0))" "((0, 1, 0), (1, 0, 0), (0, 0, 1))`

`((0, 1, 0), (0, 0, 1), (1, 0, 0))" "((0, 0, 1), (1, 0, 0), (0, 1, 0))" "((0, 0, 1), (0, 1, 0), (1, 0, 0))`

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 xx 6 = 36` non-singular `3xx3` matrices with `4` entries being `1` and the remaining `5` entries `0`.