Chapter 1.3, Problem 6E

In Exercises 5 and 6, assume that the given system is consistent. For each system determine, in the notation of Theorem 3, all possibilities for the number, $r$ of nonzero rows and the number, $n-r,$ of unconstrained variables. Can the system have a unique solution?

$a_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3+a_{14}{x}_4=b_1$

$a_{21}{x}_1+a_{22}{x}_2+a_{23}{x}_3+a_{24}{x}_4=b_2$

$a_{31}{x}_1+a_{32}{x}_2+a_{33}{x}_3+a_{34}{x}_4=b_3$