If `a , b , c` are non-zero, then the system of equations `(alpha+a)x+alphay+alphaz=0,alphax+(alpha+b)y+alphaz=0,alphax+alphay+(alpha+c)z=0` has a non-trivial solution if `alpha^(-1)=-(a^(-1)+b^(-1)+c^(-1))` b. `alpha^(-1)=a+b+c` c.`alpha+a+b+c=1` d. none of these

Options

A. `2alpha=a+b+c`
B. `alpha^(-1)=a+b+c`
C. `alpha+a+b+c=1`
D. `alpha^(-1)=-(a^(-1)+b^(-1)+c^(-1))`

Answer

D

Explanation

`(d)` The given system of equations will have a non-trivial solution if
`|{:(alpha+a,alpha,alpha),(alpha,alpha+b,alpha),(alpha,alpha,alpha+c):}|=0`
Operate `R_(2)toR_(2)-R_(1)` , `R_(3)-R_(3)-R_(1)`, then
`|{:(alpha+a,alpha,alpha),(-a,b,0),(-a,0,c):}|=0`
`impliesalpha(bc+ca+ab)+abc=0`
Since, `a,b,c ne 0`
`:.(1)/(alpha)=-((1)/(a)+(1)/(b)+(1)/(c ))`