Activity 1.6.1.
The linear system
\begin{equation*}
\left\{\begin{array}{l}
a_{11} x + a_{12} y = b_1\\
a_{21} x + a_{22} y = b_2
\end{array}\right.
\end{equation*}
is called general form of a linear system with two variables and two equations, where \(a_{ij}, b_{i}\) are arbitrary real numbers. It is claimed that if
\begin{equation*}
a_{11}a_{22}-a_{12}a_{21}\neq 0,
\end{equation*}
then the linear system is always consistent. The proof is very lengthy, which involves in a lot of case-by-case discussion. We are not going to prove the claim here. Instead, we are going to see where the value \(a_{11}a_{22}-a_{12}a_{21}\neq 0\) hides in the row-echelon form of the linear system.
Just for the purpose of avoiding very tedious discussion, we may assume that \(a_{11}\neq 0\) (why we may assume that? answer it from the viewpoint of equivalent linear systems). As you may expected, this number must appear in the row echelon form of the augmented matrix. Let’s begin.
Did you see where the number hides? try simplify the entry
A[2,2]
There is a better way to see the matrix \(A\text{.}\)
