MHB Construct 3 Augmented Matrices for Linear Systems

[karush]

$\tiny{311.1.1.26}$
Construct 3 augmented matrices for linear systems whose solution set is $x_1=3, \quad x_2=-2, \quad x_3=-1$
ok the only thing I could think of is just rearrange the rows of an RREF matrix. albeit losing the triangle format
hopefully no typos
$\left[\begin{array}{rrr|rr}
1& 0& 0& 3\\ 0& 1& 0& -2\\ 0& 0& 1& -1\\
\end{array}\right]
\quad
\left[\begin{array}{rrr|rr}
0& 1& 0& -2\\ 1& 0& 0& 3\\ 0& 0& 1& -1\\
\end{array}\right]
\quad
\left[\begin{array}{rrr|rr}
0& 1& 0& -2\\0& 0& 1& -1\\ 1& 0& 0& 3\\
\end{array}\right]
$

 

[HOI]

If the problem only asks for "3 augmented matrices for linear systems whose solution set is $x_1=3$, $x_2=−2$, $x_3=−1$" then that is perfectly good. Your teacher might complain that you are not being creative enough but that is the fault of the question! You can get other matrices by "row operations" on those matrices, basically the reverse of using row operations to solve matrix equations.

For example, adding 3 times the second row of the first of your three matrices to the first row gives
$\left[\begin{array}{rrr|rr}
1& 3& 0& -3\\ 0& 1& 0& -2\\ 0& 0& 1& -1\\
\end{array}\right]
$

 

[karush]

ok that helps a lot i will try to get back to this tmro
btw you have really helped me a lot in this forum, and deeply appreciate it
I'm retired so pretty much on my own except for the zoom classes