I am new to this so please don't make fun of me. The question:
Suppose that the linear system
$$
\begin{align*}
2x + 4y &= f \\
cx + dy &= g
\end{align*}
$$
is consistent for all possible values of $f$ and $g$. What can you say about
the coefficients $c$ and $d$? (Hint: What does row reduction tell you?)
I learned that this would be the augmented matrix:
$$
\begin{bmatrix}
2 & 4 & f \\
c & d & g
\end{bmatrix}
$$
So with row reduction:
Each row has to have a leading coefficient of $1$ and where the leading coefficient is $1$ then the the rest in the column has to be zeros if the leading coefficient is $1$.
Am I on the right track?
Best Answer
If I'm following you correctly, the answer is yes. You can determine when your system always has a solution by examining the row reduction procedure.
You need to figure out when row reduction will give you the proper form in the augmented matrix (the conditions that you specified in your last paragraph).
If $c$ and $d$ are both zero, the system does not always have a solution; in particular, there would be no solution when $g\ne0$.
Otherwise, let's row reduce: $$ \left[\matrix{2&4 \ \cr c & d }\Biggl| \matrix{\ f\cr\ g}\right]\rightarrow\quad \left[\matrix{1&2 \cr c\ & d \ }\Biggl| \matrix{\ f/2\cr\ g}\right] \rightarrow\quad \left[\matrix{1&2\ \cr 0 & d-2c \ }\Biggl| \matrix{\ f/2\cr\ {g }-{cf/2 }}\right] $$ The row reduction can be completed if and only if $d-2c\ne0$.
If $d-2c=0$, then by selecting certain values of $f$ and $g$, we would have a system with no solution (it would have an equation of the form $0= g-cf/2\ne0$)
If $d-2c\ne0$, then we can complete the reduction: $$ \rightarrow\quad \left[\matrix{1&2& \ \cr 0 & 1\vphantom{1\over2}\ } \Biggl| \matrix{\ f/2\cr {g }-{cf/2 }\over d-2c }\right] $$ and we can solve regardless of the values of $f$ and $g$.
So, the system always has a solution if and only if $d-2c\ne0$ (note this condition includes the case we previously considered: $c=d=0$).
As it turns out, the condition is that the first two entries of the second row do not both become 0 in the reduction. This happens if and only if the row $[ c\ d]$ is not a multiple of the row $[2\ 4]$.