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Matriks dan Ruang Vektor : Gaussian Elimination dan Metode Operasi Baris Elementer Tereduksi ( Reduced Row Echelon Form ) dalam Matriks

Metode Operasi Baris Elementer Tereduksi ( Reduced Row Echelon Form )

Properties :

1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leading 1
2. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix
3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row
4. Each column that contains a leading 1 has zeros everywhere else in that column

A matrix that has the first three properties is said to be in row echelon form

Example 1

Row Echelon Form :

Reduced Row Echelon Form :

Exercise 1

Determine whether the matrix is in row echelon form, reduced row echelon form, or neither :

Gauss-Jordan Elimination

The procedure for reducing matrix to row echelon form is called Gauss elimination. The procedure for reducing matrix to reduced row echelon form is called Gauss-Jordan elimination. Basically, Gauss-Jordan Elimination is just Gauss Elimination but instead changing all element into reduced row echelon form or MEBT ( Matriks Elementary Baris Tereduksi )--look other post to re-learn about MEBT Form--RRE form.

Read More : Matriks Dan Ruang Vektor : Sistem Persamaan Liniear Dan Representasi Matriks

Exercise 2

Solve the linear system by Gaussian elimination :

Solve the linear system in 1,2, and 3 by Gauss-Jordan elimination

Homogeneous System of Linear Equations

The system has the form as shown below :

Every homogeneous system of linear equations is consistent (trivial solution). If there are other solution, that's called nontrivial solutions

Use Gauss-Jordan elimination to solve the homogeneous linear system--which has 0 as a result of each equation :

The Zero Theorem

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If a homogeneous linear system has n unknowns, and if the reduced row echelon form of its augmented matrix has r nonzero rows, then the system has n-r free variables

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Exercise 3

Use Gaussian elimination or Gauss-Jordan elimination to solve the homogeneous linear system :