Solving a Matrix: Finding Positive & Negative Solutions

[helpm3pl3ase]

have a matrix:

X1+X2+X3+X4 = 0
X1+X2-X3-X4 = 0
X1-X2+X3-X4 = 0

the solutions that are formed are:
t
-t
-t
t

Now i understand that they just used t to label the free variables.. but Iam not sure how they were able to tell which were negative and positive.. I tried to do it out, but just confused myself more.. Help anyone?

 

[wildman]

The answer means:
X1 = t
X2 = -t
X3 = -t
X4 = t

The reason for the signs is they have to add to zero in all three cases.

For instance:
Take X1+X2+X3+X4

If you substitute the answer in you get:

t+ (-t) + (-t)+t

t+(-t) is zero and (-t)+t is zero so they add to zero.

In the second case:

X1+X2-X3-X4

In this case substituting in the answer:

t + (-t) - (-t) - t

t + (-t) is zero and -(-t)-t is zero

The third case is left as an exercise for the student.