Chapter 1.1, Problem 38E

Consider the $\left(2\times 2\right)$ system

$a_{11}{x}_1+a_{12}{x}_2=b_1$

$a_{21}{x}_1+a_{22}{x}_2=b_2$.

Show that if $a_{11}{a}_{22}-a_{12}{a}_{21}\ne 0$, then this system is equivalent to a system of the form

$c_{11}{x}_1+c_{12}{x}_2=d_1$

$c_{22}{x}_2=d_2$,

Where $c_{11}\ne 0$ and $c_{22}\ne 0$. Note that the second system always has a solution. [Hint: First suppose that $a_{11}\ne 0$, and then consider the special case in which $a_{11}=0$.]