Example 3.
x - y + 2z = 1
y + z = -2
x -3y = 5
| > |
B:=<<1,0,1>|<-1,1,-3>|<2,1,0>>; |
| > |
GaussianElimination(B); |
| > |
GaussianElimination(Bc); |
 |
(2.1.1) |
| > |
x:=BackwardSubstitute(%%); |
We see that the rank of the matrix is two and the rank of the augmented matrix is also two and we have an infinite number of solutions. After this chapter we will be using the rank command first introduced in this example to determine the rank of matrices. For this section you can use it as a check but you need to understand the idea of rank so I will want you to use Gauss elimination in the exercises.
We summarize the above results in a theorem.
Theorem 1.86
The following are true concerning the system Ax = b and the nxn coefficient matrix A.
i) A is invertible (equivalently A has a unique inverse) if and only if the above system has a unique solution (A is nonsingular). The unique solution is given by x =A^(-1)b.
ii) A is invertible if and only if det(A) is not zero.
iii) A is invertible if and only if rank (A) = n.
iv) A is invertible if and only if A has n pivots.
v) The system has a solution if and only if rank(A) = rank([Ab]). This solution is unique if and only if rank(A) = rank([Ab]) = n.
Homogeneous Systems
If Ax = 0 i.e. is called a system homogeneous system of equations. It A is nonsingular then the system as a unique solution which is given by x=0. If A is singular then the solution will be of the form .
=
+...
. where k<n. The complete solution to a system of equation will always be of the form
If A is nonsingular then 
and we have only 
.
We will now look at the above examples in light of this new information.
Here we have only the particular solution and the homogeneous part will be the zero vector. For the matrix B we will have both a particular and a homogeneous part.
The homogeneous part is the coefficients of the paramenter _t0[3] we do not need to parameter. i.e.
This is what we expected. We get only te zero solution where.
The general solution is thus xp + k* xh where k is not the parameter.
