Gauss-Jordan Matrix Elimination
Gauss-Jordan Matrix Elimination
-This method can be used to solve systems of linear equations involving two or more variables. However, the system must be changed to an augmented matrix. -This method can also be used to find the inverse of a 2x2 matrix or larger matrices, 3x3, 4x4 etc. Note: The matrix must be a square matrix in order to find its inverse. An Augmented Matrix is used to solve a system of linear equations. a1 x + b1 y + c1 z = d1 a 2 x + b2 y + c 2 z = d 2 a3 x + b3 y + c3 z = d 3
System of Equations ⎯ ⎯→
Augmented Matrix ⎯ ⎯→
⎡ a1 ⎢ ⎢a 2 ⎢ a3 ⎣
b1 b2 b3
c1 d1 ⎤ ⎥ c2 d 2 ⎥ c3 d 3 ⎥ ⎦
-When given a system of equations, to write in augmented matrix form, the coefficients of each variable must be taken and put in a matrix. For example, for the following system:
3x + 2 y − z = 3 x − y + 2z = 4 2x + 3 y − z = 3
Augmented Matrix ⎯ ⎯→
⎡ 3 2 − 1 3⎤ ⎢ ⎥ ⎢1 − 1 2 4 ⎥ ⎢ 2 3 − 1 3⎥ ⎣ ⎦
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-There are three different operations known as Elementary Row Operations used when solving or reducing a matrix, using Gauss-Jordan elimination method. 1. Interchanging two rows. 2. Add one row to another row, or multiply one row first and then adding it to another. 3. Multiplying a row by any constant greater than zero.
Identity Matrix-is the final result obtained when a matrix is reduced. This matrix consists of ones in the diagonal starting with the first number.
-The numbers in the last column are the answers to the system of equations. ⎡1 0 0 3 ⎤ ⎢ ⎥ ⎯⎯ ⎢0 1 0 2⎥ ← Identity Matrix for a 3x3 ⎢0 0 1 5 ⎥ ⎣ ⎦
⎡1 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎣ 0 0 0 2⎤ ⎥ 1 0 0 6⎥ ← Identity Matrix for a 4x4 ⎯⎯ 0 1 0 1⎥ ⎥ 0 0 1 4⎥ ⎦
-The pattern continues for bigger matrices.
Solving a system using Gauss-Jordan
–The best way to go is to get the ones first in their respective column, and then using that one to get the zeros in that column. -It is very important to understand that
-This method can be used to solve systems of linear equations involving two or more variables. However, the system must be changed to an augmented matrix. -This method can also be used to find the inverse of a 2x2 matrix or larger matrices, 3x3, 4x4 etc. Note: The matrix must be a square matrix in order to find its inverse. An Augmented Matrix is used to solve a system of linear equations. a1 x + b1 y + c1 z = d1 a 2 x + b2 y + c 2 z = d 2 a3 x + b3 y + c3 z = d 3
System of Equations ⎯ ⎯→
Augmented Matrix ⎯ ⎯→
⎡ a1 ⎢ ⎢a 2 ⎢ a3 ⎣
b1 b2 b3
c1 d1 ⎤ ⎥ c2 d 2 ⎥ c3 d 3 ⎥ ⎦
-When given a system of equations, to write in augmented matrix form, the coefficients of each variable must be taken and put in a matrix. For example, for the following system:
3x + 2 y − z = 3 x − y + 2z = 4 2x + 3 y − z = 3
Augmented Matrix ⎯ ⎯→
⎡ 3 2 − 1 3⎤ ⎢ ⎥ ⎢1 − 1 2 4 ⎥ ⎢ 2 3 − 1 3⎥ ⎣ ⎦
The Math Center
■
Valle Verde
■
Tutorial Support Services
■
EPCC
1
-There are three different operations known as Elementary Row Operations used when solving or reducing a matrix, using Gauss-Jordan elimination method. 1. Interchanging two rows. 2. Add one row to another row, or multiply one row first and then adding it to another. 3. Multiplying a row by any constant greater than zero.
Identity Matrix-is the final result obtained when a matrix is reduced. This matrix consists of ones in the diagonal starting with the first number.
-The numbers in the last column are the answers to the system of equations. ⎡1 0 0 3 ⎤ ⎢ ⎥ ⎯⎯ ⎢0 1 0 2⎥ ← Identity Matrix for a 3x3 ⎢0 0 1 5 ⎥ ⎣ ⎦
⎡1 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎣ 0 0 0 2⎤ ⎥ 1 0 0 6⎥ ← Identity Matrix for a 4x4 ⎯⎯ 0 1 0 1⎥ ⎥ 0 0 1 4⎥ ⎦
-The pattern continues for bigger matrices.
Solving a system using Gauss-Jordan
–The best way to go is to get the ones first in their respective column, and then using that one to get the zeros in that column. -It is very important to understand that