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)=`
A
`2^(n)`
B
`2^(n-1)`
C
`2^(2n)`
D
`2^((n-1)^(2))`
Text Solution
Verified by Experts
The correct Answer is:
D
`(d)` 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.
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