What Does ##ad-bc=0## Imply About Solutions to ##AX=0##?

[Bashyboy]

Homework Statement

Consider the system $AX=0$, where
$$A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$
is a 2x2 matrix over the field F. Prove that if $ad-bc=0$ and some entry of $A$ is different from $0$, then there is a solution $(p,q)$ such that $(x,y)$ is a solution if and only if there is some scalar $t$ such that $x=pt$ and $y = qt$.

Homework Equations

The Attempt at a Solution

Am I asked to find the vector $(p,q)$ for which the statement "$(x,y)$ is a solution if and only if there is some scalar $t$ such that $x=pt$ and $y = qt$" holds, or am I assuming that I have such a solution $(p,q)$ and proving that this statement?

 

[stevendaryl]

Am I asked to find the vector $(p,q)$ for which the statement "$(x,y)$ is a solution if and only if there is some scalar $t$ such that $x=pt$ and $y = qt$" holds, or am I assuming that I have such a solution $(p,q)$ and proving that this statement?

Well, the problem doesn't ask you to find $(p,q)$, only to show that if $(p,q)$ is a solution, and $(x,y)$ is a second solution, then there is some $t$ such that $x=pt$ and $y=qt$.

For a 2x2 matrix, it's easy enough to convert $\left( \begin{array}\\ a & b \\ c & d \end{array} \right) \left( \begin{array} \\ x \\ y \end{array} \right) = 0$ into two ordinary (non-matrix) equations for $x$ and $y$ to see how the possible values of $x$ and $y$ must be related.