Gaussian Elimination

For an equation of the form $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$, the set of values for $\boldsymbol{x}$ that satisfies the equation is the solution for this System of Linear Equations.

Elementary Transformations

For a given set of linear equations, a certain set of operations over the rows that change its form while retaining the set of solutions:

• Exchange two rows.
• Multiply a row with a constant $\lambda \in \mathbb{R} \setminus \{0\}$.
• Addition of two rows.

(Row-Echelon Form)

A matrix is in row-echelon form (REF) if:

• All rows that contain only zeros are at the bottom of the matrix.
• Looking at the nonzero rows only, the first nonzero number from the left (pivot) is always strictly to the right of the pivot of the row above it.

(Pivot and Staircase Structure)

The leading coefficient of a row (first non-zero number from the left) is called a pivot and is always strictly to the right of the row above it.

(Basic and Free variables). The variables corresponding to the pivots in the REF are called basic variables and the other variables are called free variables. (tip: think about the column view).

(Reduced Row-Echelon Form)

A system of equations is in reduced row echelon form if:

• It is in REF.
• Every pivot is 1.
• The pivot is the only nonzero entry in its column.

(Gaussian Elimination)

Gaussian elimination is an algorithm that elimination performs elementary transformations to bring a system of linear equations into RREF.

(Finding the inverse)

Notice that the Gaussian Elimination could be used to find the inverse of a square matrix too.

$$\begin{align} \begin{bmatrix} A|I_{n} \end{bmatrix}\rightsquigarrow\dots\rightsquigarrow \begin{bmatrix} I_{n}|A^{-1} \end{bmatrix} \tag{1} \end{align}$$

Connections

• Forward Links:
• Backward Links: Matrix, Inverse and Transpose, System of Linear Equations