Lat`A = [a_(ij)]_(3xx 3).` If tr is arithmetic mean of elements of rth row
and `a_(ij )+ a_( jk) + a_(ki)=0` holde for all `1 le i, j, k le 3.`
Matrix A is

Options

A. non- singular
B. symmetric
C. skew-symmetric
D. nether symmetric nor skew-symmetric

Answer

C

Explanation

`therefore A = [[a_(11) , a_(12),a_(13)],[a_(21),a_(22), a_(23) ],[a_(31), a_(32),a_(33)]]`
`rArr t_(1) = (a_(11) + a_(12)+a_(23))/3 = 0, [because a_(ij) + a_(jk) + a_(ki)=0]`
`t_(2) = (a_(21) + a_(22) + a_(23))/3 = 0`
and `t_(3) = (a_(31) + a_(32) + a_(33))/3 = 0`
`because a_(11) + a_(11) + a_(11) = 0, a_(11) + a_(12) + a_(21)= 0,`
` a_(11) + a_(13) + a_(31) = 0, a_(22) + a_(22) + a_(22)= 0, `
` a_(22) + a_(12) + a_(21) = 0, a_(22) + a_(22) + a_(22)= 0, `
` a_(33) + a_(13) + a_(31) = 0, a_(33) + a_(23) + a_(32)= 0, `
and ` a_(33) + a_(12) + a_(21) = 0,` we get
`a_(11) = a_(22) = a_(33) = 0 `
and `a_(12) =-a_(21), a_(23) = - a_(32), a_(13) = -a_(31)`
Hence, A is skew - symmetric matrix.