$ \begin{array}{ccc}{1}&{\mathrm{{-}}{4}}&{{7}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}}&{g}\\{0}&{3}&{\mathrm{{-}}{5}}&{h}\\{\mathrm{{-}}{2}}&{5}&{\mathrm{{-}}{9}\hspace{0.33em}\hspace{0.33em}}&{k}\end{array} $
Shown above is augmented matrices and the question is
Find an equation involving g,h,k that makes the augmented matrix correspond to a consistent system.
After R2/3 , R3+2R1 and R3+R2, I got
$1 \hspace{5pt}-4 \hspace{10pt} 7 \hspace{10pt} g \\$
$0 \hspace{5pt}1 \hspace{5pt} -5/3 \hspace{10pt} h/3 \\$
$0\hspace{5pt}0 \hspace{5pt} 1 \hspace{5pt} (h+k+2g)/7 \\$
Which tells me that h,k,g can have any value for the system to be consistent, but the answer given is h+k+2g=0
I feel it should be h+k+2g=a
$\endgroup$ 31 Answer
$\begingroup$First, reduce the given matrix into echelon form. Then the last row will be in the form $$\begin{bmatrix}0 & 0 & 0 & k+2g+h\end{bmatrix}.$$ For a matrix to be consistent, all the elements in the last row of the matrix should be zero. Therefore, $k+2g+h=0$ is the answer.
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