A Set of Linear Equations Equal to Zero

Question

If A+B+C+D = 1;

Let A,B,C,D be elements of vector V.

Let there be a 4*4 matrix M as such :

  -5  5  0  0
   4 -8  4  0
   0  5 -7  2
   0  0 10 -10

A property exist that VM = 0.

• What is V=[?, ? ,? ,?] ?

I have tried to solve this on matlab and failed. I always reach to conclusion that the vector V are all 0,0,0,0.

Also that solves the equation as well, that is not the answer i am looking for since the SUM of V=1.

The answer should be a non zero value.

These are the set of equations:

4b-5a =0
5a-8b+5c=0
4b-7c+10d=0
2c-10d=0

Answer 1

Are you sure that $vM=0$ has infinitely many solutions for $v$?

The question asked is more of a matlab question than a math question: you need to find the function that returns the space of all solutions (i.e. the left nullspace of $M$), not just one.

Or, you could ask matlab to solve the equation you're really trying to solve:

  /  |1\
v |M |1| = [0 1]
  |  |1|
  \  |1/

Answer 2

Add the first and second equation. This gives $-4b = 0$, $b = 0$, and subsequently using the first equation $a = 0$.

Adding the third and fourth equation gives $c = 0$, $d = 0$.

Therefore $Mv$ does note have non-trivial solutions.