Showing a solution for system of equations

Plug in $(ac_1,ac_2)$ ...
$<br /> A_{11}(ac_1)+A_{12}(ac_2)$
$=a\left[A_{11}(c_1)+A_{12}(c_2)\right]$
$=a[0]$
$=0$

Therefore, if $A_{11}(c_1)+A_{12}(c_2)=0$ , then $(ac_1,ac_2)$ is also a solution to $A_{11}x_1+A_{12}x_2=0$ .

then repeat the process of the second equation. This process can be done more quickly if you use matrix notation:

$\vec{c}$ is a solution to $A\vec{x}=\vec{0}$
$\rightarrow A\vec{c}=\vec{0}$

so, if we test $a\vec{c}$ ...
$A(a\vec{c})=aA\vec{c}=a\vec{0}=\vec{0}$

therefore $a\vec{c}$ is also a solution to $A\vec{x}=\vec{0}$