For what value of $k$ the system
$
\left\{
\begin{array}{c}
kx+ay=5 \\
ax+ky=k \\
\end{array}
\right.
$
has infinite solutions?
Honestly, i don't know how to start this problem, i saw that this have something to do with matrices and the determinants, but this is meant to be done without that.
The possible answers are:
$A) $ $-5$ or $5$
$B) \sqrt 5$ or $-\sqrt 5$
$C) -25$ or $25$
$D) 0$
$E)$ Can't be determinated
Best Answer
Since both the equations have infinite solutions, they must coincide with each other
This gives:
$\frac{k}{a}$ = $\frac{a}{k}$ = $\frac{5}{k}$
Considering the equation $\frac{a}{k}$ = $\frac{5}{k}$ and solving it, $a$ turns out to be $5$.
Considering the equation $\frac{k}{a}$ = $\frac{a}{k}$ and knowing that $a = 5$ and solving it, $k^2$ turns out to be 25.
This means $k$ can be $5$ or $-5$.
Hence option $A$ is right.