I am having difficulty with true/false statements and their justifications regarding systems of linear equations.
(a) A linear system of three equations in five unknowns is always consistent (i.e. it has at least one solution)
(b) A linear system of five equations in three unknowns cannot be consistent
(c) If a linear system in echelon form is triangular then the system has the unique solution
(d) If a linear system of n equations in n unknowns has two equations that are multiples of one another,
then the system is inconsistent.
So far, for (a) I have said False, as it will always be consistent if it is homogeneous, but not if it is non-homogeneous.
For (b) I have said false, but am having difficulty justifying this assertion
(c) I know to be true.
(d) I believe may be false as having equations that are multiples could result in free variables and hence infinite solutions?
I am rather unsure on what I have done so far.
Any assistance is greatly appreciated.
Best Answer
for part $b$ , you may consider below counter example :
$x_1+x_2+x_3=1$
$x_1+x_2+x_3=1$
$x_1+x_2+x_3=1$
$x_1+x_2+x_3=1$
$x_1+x_2+x_3=1$
for part $c$ , look @ Is it possible for a triangular matrix in echelon form to not have a unique solution and how?
for part $d$ :
you're correct about the answer but your reasoning is only partially correct - the system can also be inconsistent :
$x_1+x_2+x_3 = 1$
$2x_1+2x_2+2x_3=2$
$x_1+x_2+x_3 = 2$