A `3xx3` determinant has entries either `1` or `-1`.
Let `S_(3)=` set of all determinants which contain determinants such that product of elements of any row or any column is `-1` For example `|{:(1,,-1,,1),(1,,1,,-1),(-1,,1,,1):}|`is an element of the set `S_(3)`.
Number of elements of the set `S_(3)=`

Options

A. `10`
B. `16`
C. `12`
D. `18`

Answer

B

Explanation

`(b)` For `S_(n),a_(11),a_(12),a_(13),….a_(1(n-1))` we have two options `'1'` or `'-1'`m but for `a_(1n)` we have only one way depending upon the product `(a_(11)*a_(12)*a_(13)*…..*a_(1(n-1)))`
`:.` For `R_(1)` we have `2^(n-1)` ways
Similarly for `R_(2),R_(3),R_(4),....R_(n-1)` we have `2^(n-1)` ways
For `R_(n)` we have only one way.
Hence total number of ways `(2^(n-1))^(n-1)=2^((n-1)^(2))`
For `S_(3)`, we have `2^((3-1)^(2))=1` elements.