Gaussian Elimination in Matrix Form

Gaussian Elimination is a systematic method for solving systems of linear equations by transforming the augmented matrix representing the system into its row-echelon form. The process involves elementary row operations, including swapping rows, multiplying a row by a constant, and adding or subtracting rows.

Steps:

1. Start with an Augmented Matrix: Represent the system of linear equations in an augmented matrix [A|B], where A is the coefficient matrix and B is the column of constants.

2. Row Reduction: Apply row operations to transform the matrix into row-echelon form, where leading entries (pivot elements) are along the main diagonal, and entries below and above pivots are zeros.

3. Analyze Rows:

   • If a row is of the form 0x + 0y + 0z = k (where k is not zero), the system is inconsistent.
   • Rows of the form 1x + 0y + 0z = k indicate one solution.
   • Rows with all zero entries imply infinitely many solutions.

4. Back Substitution (Optional): If the matrix is square and in row-echelon form, perform back substitution to find the values of variables.

Gaussian Elimination simplifies solving linear systems, providing insight into the existence and nature of solutions.

Gaussian Elimination Example

Consider the system of equations:

$2x + y - z = 8$
$-3x - y + 2z = -11$
$-2x + y + 2z = -3$

Representing this system as an augmented matrix [A|B]:

$\left[\begin{array}{ccc|c} 2 & 1 & -1 & 8 \\ -3 & -1 & 2 & -11 \\ -2 & 1 & 2 & -3 \end{array}\right]$

Row Reduction:

1. Row 2 = Row 2 + (3)Row 1

$\left[\begin{array}{ccc|c} 2 & 1 & -1 & 8 \\ 0 & 2 & -1 & 5 \\ -2 & 1 & 2 & -3 \end{array}\right]$

2. Row 3 = Row 3 + Row 1

$\left[\begin{array}{ccc|c} 2 & 1 & -1 & 8 \\ 0 & 2 & -1 & 5 \\ 0 & 2 & 1 & 5 \end{array}\right]$

3. Row 3 = Row 3 - Row 2

$\left[\begin{array}{ccc|c} 2 & 1 & -1 & 8 \\ 0 & 2 & -1 & 5 \\ 0 & 0 & 2 & 0 \end{array}\right]$

Back Substitution:

Solving for variables gives (z = 0), (y = 2), and (x = 3). Therefore, the system has a unique solution.

Gaussian Elimination simplifies solving linear systems, providing a clear path to solutions.

Numbers of solutions

Compute the rank:

the rank of a matrix is the deminsion of the vector space generated or spanned by its columns. The rank is therefore the number of linear independent columns of our matrix. This, in turn, is identical to the dimension of the vector space spanned by its rows.

$rank(A):= dim(img(A))$

TO compute the rank of a matrix use gauss elemination and count the non-zero rows of the transformed matrix.

A system of linear equations is only solvable if $rank(A) = rank(A|B)$

1. unique solution: $rank(A) = n$

2. infinitely solutions: $rank(A) \neq n$