Linear equations:

The system of homogeneous simultaneous linear relations

      has a non-trivial solution (i.e. at least one of x,y,z, is different from zero ) if

If Δ ≠ 0, then the only solution of the over system of equations is x = 0, y = 0,  z = 0 (known as trivial solution).

If  Δ = 0, then the number of solutions of the system of equations is infinite (known as non-trivial and adds trivial solution).

Example: If the system of equations 3x + 10 y + 17z =  0 ,  -x + 6 y + 13 z = 0 and  20 x - 13y + l z = 0 has a non-trivial solution then calculate the solution.

Solution:       The observation for A.P. property reveals l = -46 to have D = 0 (non-trivial solution)

                        Suppose  z = k  and  by   first two relations, we have  3x +10 y = - 17 k - 3x + 18y  = - 39 k, therefore 

                          y = - 2k and  x = k 

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