Question

(b) Find all 3-tuples (a,b,c) satisfying

Answer 1

We have a three unknown, 4 equation homogeneous system.  These always have at least (0,0,0) as a solution.  Let's write the equations, one column at a time.

1a + 0b + 0c = 0

-1a + 1b +0c = 0

0a - 1b + c = 0

0a + 0b + -1 c = 0

We could do row reduction but these are easy enough not to bother.

Equation 1 says

a = 0

Equation 4 says

c = 0

Substituting in the two remaining,

-1(0) + 1b + 0c = 0

b = 0

0(0) - 1b + 0 = 0

b = 0

The only 3-tuple satisfying the vector equation is (a,b,c)=(0,0,0)