Given the augmented matrix of a linear system : $$ \begin{bmatrix} 1 & * & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \\ \end{bmatrix} $$ with * are unknown real numbers. Determine whether the system has unique solution, many solutions, or has no solution.
Attempt :
The answer key from the textbook says it has unique solution. But, if the $* = 0$, we get a counterexample
$$ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} $$
Which means that $a=c_{1}, b=c_{2}, c = c_{3}$, but the $d$ is free.
Notice that in general we can do operations so that it will become
$$ \begin{bmatrix} 1 & 0 & 0 & * \\ 0 & 1 & 0 & * \\ 0 & 0 & 1 & * \\ \end{bmatrix} $$
So again, whether * nonzero or not, we will have infinitely many solutions.
How is this analysis? I presume the answer key is mistaken. The textbook is Elementary Linear Algebra applications version 11, by Howard Anton and Chris Rorres. Thanks.
If it is an augmented system, isn't the last column the right-hand side? In that case, the augmented system is upper triangular, with ones on the diagonal. What does that tell you?