I am working on problem #25 of Linear Algebra and its Applications and the question asks:
Find an equation involving $g$, $h$, and $k$ that makes this augmented matrix correspond to a consistent system:
$$\left(\begin{array}{ccc|c}
1& -4& 7& g \\
0& 3& -5& h \\
-2& 5& -9& k
\end{array}\right).$$
After I do $R_3 \gets 2R_1 + R_3$ and $R_3 \gets R_2 + R_3$
I end up with
$$\left(\begin{array}{ccc|c} 1& -4& 7& g \\ 0& 3& -5& h \\ 0& 0& 0& 2g+k+h
\end{array}\right).$$
For this to be a consistent system the third row should be $\begin{pmatrix}0& 0& 0& 0\end{pmatrix}$,
so in order for this augmented matrix to be a consistent system then $2g + k + h =0$
The answer in the back of the book is $k – 2g + k = 0$.
Where am I going wrong with my calculation? Or is the book wrong?
Best Answer
Just for the record, the 3rd edition of Lay's book has your answer to this question.