1. ## Augmented Matrix Problem

Hello, I have tried this problem out but get stuck on one part.

Find an equation involving g, h and k that makes this augmented matrix consistent.

1 -4 7 g
0 3 -5 h
-2 5 -9 k

so i try and put this in trangular form by multiplying row_1 by 2 and adding it to row_3 and end up with
1 -4 7 g
0 3 -5 h
0 -3 5 k
can i assume since row 2 and row 3 are almost the same, just scaled by -1 that i can take out one of the equations since they are the same? would column 3 be a free variable?

The 'k' in the second tableau should be "k + 2g"

3. ahhh yes, thanks, i forgot to look at the right side of the matrix.

4. Hello, carlo_b!

Find an equation involving g, h and k that makes this augmented matrix consistent.

. . $\begin{pmatrix}1 & \text{-}4 & 7 &|& g \\
0 & 3 & \text{-}5 &|& h \\
\text{-}2 & 5 & \text{-}9 &|& k \end{pmatrix}$

Your next step is a bit off . . .

$\begin{array}{c} \\ \\ R_3 + 2\!\cdot\!R_1\end{array}\;\begin{pmatrix}1 & \text{-}4 & 7 &|& g \\
0 & 3 & \text{-}5 &|& h \\
0 & \text{-}3 & 5 &|& {\color{blue}k+2g} \end{pmatrix}$

Can i assume since row 2 and row 3 are almost the same, just scaled by -1,
that i can take out one of the equations since they are the same? . . Yes
Would column 3 be a free variable? . Yes
One more step . . .

$\begin{array}{c}\\ \\ \text{-}1\!\cdot\!R_3\end{array}\;\begin{pmatrix}1 & \text{-}4 & 7 &|& g \\ 0 & 3 & \text{-}5 &|& h \\ 0 & 3 & \text{-}5 &|& \text{-}2g-k\end{pmatrix}$

If $h$ is not equal to $\text{-}2g-k$, the system is inconsistent.

To be consistent: . $h \;=\;\text{-}2g-k$

Ahh, too slow again . . .