Augmented Matrix Algebra Problem

It is $\color{#CEBB00}{\boxed{false}}$ .

$\color{#D61F06}{Ax = b}$ represents a linear system where the vector $x$ is being operated on by matrix $A$ and then is equal to vector $b$ .

When you have a system of equations you can represent it in this form where $A$ contains the coefficients, $x$ contains your variables and $b$ contains the values on the other side of the equal sign.

When we solve a system of equations via Gauss-Jordan elimination we are doing row operations to both sides of the linear system. If I add $3$ times row $2$ to row $4$ for example it is happening to both the $Ax$ and the $b$ .

Let $\color{#3D99F6}{E_i}$ be elementary row operation matrices

$\color{#3D99F6}{Ax = b \Rightarrow E_1 Ax = E_1 b \Rightarrow E_2 E_1 Ax = E_2 E_1 b \Rightarrow ... }$

We just continue doing these operations until we have a reduced row echelon form, or some other easy to work with form.

An augmented matrix is what we get when we attach the vector $b$ to the righthand side of $A$ , typically with a dividing vertical line. The $x$ being implicit in our problem. This way we are free to do row operations on only one object instead of $2$ .