Consider a matrix `A=[a_("ij")]` of order `3xx3` such that `a_("ij")=(k)^(i+j)` where `k in I`.
Match List I with List II and select the correct answer using the codes given below the lists.

Options

A. `{:(a,b,c,d),(r,p,s,q):}`
B. `{:(a,b,c,d),(s,p,q,r):}`
C. `{:(a,b,c,d),(r,p,q,s):}`
D. `{:(a,b,c,d),(q,p,r,s):}`

Answer

C

Explanation

Let `A=[A_("ij")]_(3xx3)`, where `a_("ij")=(k)^(i+j)`
So, `A=[(k^(2),k^(3),k^(4)),(k^(3),k^(4),k^(5)),(k^(4),k^(5),k^(6))]`
a. If A is singular, then `|A|=0`
`implies k^(2).k^(3).k^(4) |(1,1,1),(k,k,k),(k^(2),k^(2),k^(2))|=0`,
`implies k in I`
b. If A is null matrix, then `k in {0}`
c. There is no value of `k` for `A` to be skew-symmetric matrix which is not null-matrix.
`:. k in phi`
d. If `A^(2)=3A`, then
`[(k^(2),k^(3),k^(4)),(k^(3),k^(4),k^(5)),(k^(4),k^(5),k^(6))][(k^(2),k^(3),k^(4)),(k^(3),k^(4),k^(5)),(k^(4),k^(5),k^(6))]=[(3k^(2),3k^(3),3k^(4)),(3k^(3),3k^(4),3k^(5)),(3k^(4),3k^(5),3k^(6))]`
`implies [(k^(4)+k^(6)+k^(8),k^(5)+k^(7)+k^(9),k^(6)+k^(8)+k^(10)),(k^(5)+k^(7)+k^(9),k^(6)+k^(8)+k^(10),k^(7)+k^(9)+k^(11)),(k^(6)+k^(8)+k^(10),k^(7)+k^(9)+k^(11),k^(8)+k^(10)+k^(12))]`
`=[(3k^(2),3k^(3),3k^(4)),(3k^(3),3k^(4),3k^(5)),(3k^(4),3k^(5),3k^(6))]`
`implies k in {-1, 0, 1}`