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  • If the system of linear equations x+2ay+az=0 x+3 by+bz=0 x+4 cy +cz=0 has a non-trivial solution , then a,b ,c are in

If the system of linear equations x+2ay+az=0 x+3 by+bz=0 x+4 cy +cz=0 has a non-trivial solution , then a,b ,c are in

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Solution:   The system of equations can be written as

                  AX=0, where  A=eginbmatrix 1& 2a &a \1 &3b &b \ 1 &4c &c endbmatrix,

             X=eginbmatrix x\ y\ z endbmatrix,   and     o=eginbmatrix 0\ 0\0 endbmatrix.

 For non-trivial solution of given system  of homogeneous equation,D=0

Rightarrow              eginvmatrix 1 &2a &a \ 1 & 3b & b\ 1& 4c &c endvmatrix=0  Applying R_1 ightarrow R_1-R_2 and R_2 ightarrow R_2-R_3, we have

                  eginvmatrix 0 &(2a-3b) &(a-b) \ 0 &(3b-4c) &(b-c) \ 1 &4c &c endvmatrix=0

 Rightarrow            ab+bc=2ac    or    b=2ac/(a+c)

 

Posted by

Deependra Verma

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