Linear Algebra - Augmented Matrix

Augmented Matrix

Here $\tilde{A}$ is a so-called augmented matrix, $\tilde{A}=\left(\begin{array}{llll}{a}_{11}& \dots & a_{1n}& b_1\\ ⋮& & ⋮& ⋮\\ a_{n1}& \dots & a_{nn}& b_n\end{array}\right)$

Consider the system of equations:

$$\begin{array}{rl}x+y+2z& =2\\ x+y+z& =3\\ 2x+2y+2z& =6.\end{array}$$

The coefficient matrix is $A=\left[\begin{array}{lll}1& 1& 2\\ 1& 1& 1\\ 2& 2& 2\end{array}\right]$, The augmented matrix is $(A\mid B)=\left[\begin{array}{llll}1& 1& 2& 2\\ 1& 1& 1& 3\\ 2& 2& 2& 6\end{array}\right]$

The $(A-\lambda I)x=0$ implies that $A-\lambda I$ must be singular matrix(non-invertible -$\det =0$), this is because that x is non-zero vector and the equation $(A-\lambda I)x=0$, therefore $\det (A-\lambda I)$​

Singular matrix is just the matrix that its det is 0, which is also called non-invertible matrix.

Row echelon Form(REF)
A matrix is in row echelon form (REF) if it satisfies the following three properties:

1. All nonzero rows are above any rows of all zeros.
2. Each leading (nonzero) entry of a row is in a column to the right of the leading (nonzero)
   entry of the row above it. The leading entry is also called Pivot
3. All entries in a column below a leading (nonzero) entry are zeros.
   Reduced Row Echelon (RREF)
   A matrix is in reduced row echelon form (RREF) if it satisfies the following three
   properties:
4. It is in REF;
5. The leading (nonzero) entry in each row is 1.
6. Each leading 1 is the only nonzero entry in its column.

Gaussian Elimination