Showing if linear system is degenerate, non-degenerate or inconsistent.

Question

Degenerate means that solution exists and is not unique. Non- degenerate means solution exists and is unique. Inconsistent means a solution doesnt exist.

For i) i said that the system is degenerate because x1 = c1, x2 = c2 and x3=λ where λ is an real number, so λ can be any number and therfore if not unique.

For ii) i said that the system is non-degenerate because x1= c1, x2= c2 and x3= c3 and c1,c2,and c3 are unique.

For iii) i said system is inconsistent because 0 = 1 is not true and is not a solution.

Am i right? and what am i missing?

Answer 1

Your answer is correct.

In general, for RREF, to check for inconsistent, we look for a row with all zeros followed by a $1$ at the right.

To check for non-uniqueness, we check for existence of non-pivot columns.

Otherwise, the solution is unique.

Answer 2

i) is fine, $x_3$ is indeed free to choose from $\mathbb{R}$, the solution set being $\{ (c_1, c_2, \lambda)^\top \mid \lambda \in \mathbb{R} \}$
ii) is fine, you gave the unique solution $x = (c_1, c_2, c_3)^\top$
iii) is fine as well, you recognized the inconsistent last equation $0 = 1$.