Consider an arbitarary `3xx3` non-singular matrix `A[a_("ij")]`. A matrix `B=[b_("ij")]` is formed such that `b_("ij")` is the sum of all the elements except `a_("ij")` in the ith row of A. Answer the following questions :
If there exists a matrix X with constant elemts such that AX=B`, then X is

Options

A. skew-symmetric
B. null matrix
C. diagonal matrix
D. none of these

Answer

D

Explanation

`A=[(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33))]`
`implies B=[(a_(12)+a_(13),a(11)+a_(13),a_(11)+a_(12)),(a_(22)+a_(23),a_(21)+a_(23),a_(21)+a_(22)),(a_(32)+a_(33),a_(31)+a_(33),a_(31)+a_(32))]`
`implies X=A^(-1) B`
`=1/(|A|)[(C_(11),C_(21),C_(31)),(C_(12),C_(22),C_(32)),(C_(13),C_(23),C_(33))]`
`[(a_(12)+a_(13),a_(11)+a_(13),a_(11)+a_(12)),(a_(22)+a_(23),a_(21)+a_(23),a_(21)+a_(22)),(a_(32)+a_(33),a_(31)+a_(33),a_(31)+a_(32))]`
`=1/(|A|) [(0,|A|,|A|),(|A|,0,|A|),(|A|,|A|,0)]=[(0,1,1),(1,0,1),(1,1,0)]`
`implies |A^(-1)B|=2`
or `|A^(-1)||B|=2`
or `|B|=2|A|`