Specified a system of equations, (1), we will have one of the three probabilities for the number of solutions.
1. No solution.
2. Accurately one solution.
3. Infinitely many solutions.
Before going on to the other section we require to see at one more situations. The system of equations in (1) is termed as a non-homogeneous system whether at least one of the bi's is not zero. If though all of the bi's are zero we identify the system as homogeneous and the system will be as,
a11 x1 + a12 x2 +................+a1n xn = 0
a21 x1 + a22 x2 +.............. +a2n xn = 0
...................
an1 x1 + an2 x2 +............... +ann xn = 0 ...................(2)
Now, notice that in the homogeneous case we are guaranteed to have the following solution.
x1 + x2+....... +xn = 0
This solution is frequently termed as the trivial solution.
The fact given above can be modified to the following, for homogeneous systems.