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Gauss Jordan Method

In mathematics, the linear algebra is an important field which studies about the vector spaces, linear system of equations and various operations on them. Matrices play a very important role in linear algebra. We know that a matrix is said to be a rectangular array of numbers or quantities.

 It consists of horizontal arrays called rows as well as vertical arrays called columns. There are several different types of matrices such as - diagonal matrices, rectangular matrices, triangular matrices, square matrices, identity matrices, symmetric and skew-symmetric matrices, invertible matrices etc.

In linear algebra, many concepts are based on the square matrix which is a matrix with same number of rows and columns. Inverse of a matrix is a concept that is applied on the square matrices. A square matrix P is said to be invertible if there exists a unique matrix Q in such a way that

PQ = QP = IThen, P and Q are called inverse of each other.

There are different methods of finding inverse of matrices. Gauss Jordan method is a very important tool in finding inverse of a matrix. Also, this method is more widely used in mathematics for the purpose of finding the solution of a system of linear equations which is defined as a set of linear equations sharing a set of solution for each equation in it. In this page, we are going to learn this method in detail.

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Definition

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The Gauss-Jordan method is also known as Gauss-Jordan elimination method. It was introduced by the mathematicians Carl Friedrich Gauss and Wilhelm Jordan, after their name it is called so. This method is very useful in solving a linear system of equations. It is a technique in which a system of linear equations is resolved by the means of matrices.

This method allows the isolation of the coefficients of a system of linear equations. In Gauss-Jordan method, given matrix can be fetched to row echelon form and made simpler. We can say that the transformation of augmented matrix of the given system into reduced row-echelon form is done by the use of of row operations. Sometimes, this method is also used for finding rank of a matrix as well as finding inverse of a square invertible matrix.In Gauss-Jordan method, elementary row operations are utilized in order to solve given system of linear equations.

There are 3 kinds of elementary row operations which are discussed below:

i) Switching of Rows:

A row can be interchanged with another row.

Ri↔Rj$R_i\leftrightarrow R_j$

ii) Row multiplication:

A non-zero number can be multiplied to every element in a row.

aRi→Ri$a{R}_i\to R_i$, where a ≠$\ne$ 0

iii) Addition of Rows:

We may replaced a row by the sum of elements of a row and a multiple of corresponding elements of another row.

Ri+aRj→Ri$R_i+a{R}_j\to R_i$, where i ≠$\ne$ j

Row Reduced Echelon Form

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A matrix can be called to be in reduced row echelon form (RREF) if the following conditions are satisfied:

1) The first nonzero element of a row must be 1 which is known as leading 1.

2) The leading 1 of a row must be positioned at the right side of the leading 1 of its previous row.

3) In case of a column containing a leading 1, all the remaining elements in that column must be 0.

4) At the bottom of the matrix, there is a rows having only zeros as the elements.

If any of the above rules are violated, the matrix is not said to be in reduced row echelon form.

Lets us have a look at the examples of RREF matrices:
⎡⎣⎢100010460⎤⎦⎥$\left[\begin{array}{ccc}1& 0& 4\\ 0& 1& 6\\ 0& 0& 0\end{array}\right]$

and

⎡⎣⎢⎢⎢100060000310−40−50⎤⎦⎥⎥⎥

Method

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The steps for solving a system of linear equations by using the Gauss-Jordan method are illustrated below:
Step 1: Write the augmented matrix for the given system of linear equations.

Step 2: Use a set of elementary row operations (as discussed above) in order to perform row reduction on this matrix until we obtain a unique reduced row echelon form.

Step 3: Once RREF is obtained, write down the system of linear equations from this form.

Step 4: In this way, we get a value corresponding to each unknown variable which would be the solution. If simple equations are obtained, then they should be solved by the methods of solving equation such as substitution method or elimination method.

Example

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An examples illustrating Gauss-Jordan method is given below:

Example: Use Gauss-Jordan method for solving following system of linear equations -
2y + z = 4
x + y + 2z = 6
2x + y + z = 7

Solution: The augmented matrix is given by:

⎡⎣⎢012211121∣∣∣∣467⎤⎦⎥$\left[\begin{array}{ccc}0& 2& 1\\ 1& 1& 2\\ 2& 1& 1\end{array}\right|\begin{array}{c}4\\ 6\\ 7\end{array}]$

Interchanging R1$R_1$ and R2$R_2$, we get

⎡⎣⎢102121211∣∣∣∣647⎤⎦⎥$\left[\begin{array}{cc}1& 2\\ 0& 2& 1\\ 2& 1& 1\end{array}\right|\begin{array}{c}6\\ 4\\ 7\end{array}]$

Performing the row operation R3→R3+(−2R1)$R_3\to R_3+(-2R_1)$

⎡⎣⎢10012−121−3∣∣∣∣64−5⎤⎦⎥$\left[\begin{array}{ccc}1& 1& 2\\ 0& 2& 1\\ 0& -1& -3\end{array}\right|\begin{array}{c}6\\ 4\\ -5\end{array}]$

Performing the following row operations R1→R1$R_1\to R_1$ + (-12  . R2$R_2$) and R3→R3$R_3\to R_3$ + (12  . r2$r_2$)

⎡⎣⎢⎢100020321−52∣∣∣∣∣44−3⎤⎦⎥$\left[\begin{array}{ccc}1& 0& \frac{3}{2}\\ 0& 2& 1\\ 0& 0& -\frac{5}{2}\end{array}\right|\begin{array}{c}4\\ 4\\ -3\end{array}]$

Performing R1→R1$R_1\to R_1$ + (35  . R3)$R_3)$ and R2$R_2$ →R2$\to R_2$ + (25  . R3$R_3$)

⎡⎣⎢10002000−52∣∣∣∣∣115145−3⎤⎦⎥⎥$\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& -\frac{5}{2}\end{array}\right|\begin{array}{c}\frac{11}{5}\\ \frac{14}{5}\\ -3\end{array}]$

Performing R2→R2$R_2\to R_2$ . 12  and R3→R3$R_3\to R_3$ . -25 

⎡⎣⎢100010001∣∣∣∣1157565⎤⎦⎥⎥⎥$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|\begin{array}{c}\frac{11}{5}\\ \frac{7}{5}\\ \frac{6}{5}\end{array}]$

Hence the solution is

x = 115 

y = 75 

z = 65