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**Is my logic correct? -- L. Algebra question**

## Homework Statement

[tex]\left[ \begin{array}{cccc} 1 & 2 & -3 & 2 \\ -1 & -2 & b & 3 \\ 0 & 0 & c & 2 \end{array} \right][/tex]

(1) For what values b and c is the system in the augmented matrix above consistent?

(2) For what values b and c does the system in the augmented matrix above have a unique solution?

## The Attempt at a Solution

I tried to simplify, and after a few row operations, I end up with:

[tex]\left[ \begin{array}{cccc} 1 & 2 & -3 & 2 \\ 0 & 0 & c(b - 3) & 5c \\ 0 & 0 & 0 & 2b - 6 - 5c \end{array} \right][/tex]

Now I'm thinking this is it: the system is inconsistent for b = 3 and c != 0, and consistent for all other real numbers b and c.

Also, the system can never have a unique solution since there are free variables.

Is my reasoning faulty?