Row transformation method to solve a set of equations. Today it’s about Row transformation method in matrix algebra to solve a set of equations.

Have a look!!

### Row transformation method to solve a set of equations

There are several ways to solve a set of equations in matrix algebra like Gaussian elimination method, inverse method, LU method and so on. Row transformation method is also one of that.

Gaussian elimination method in 3 × 3 matrices

Gaussian elimination method in 4 × 4 matrices

Here I’ll explain how to use row transformation method to solve a set of equations.

Suppose I have a set of equations like

Now I have to solve these equations using row transformation method.

First of all, I’ll write the set of equations in matrix form.

Thus it will be

Next I’ll intorduce the unit matrix as a coefficient of

Therefore it will be

So the combined coefficient matrix is

Next, the task is to use elementary row operations in such a way that the unit matrix comes on the left-hand side of the coefficient matrix like

Now elementary row operations mean

- interchanging rows.
- adding or subtracting a constant number to each element of a row.
- multiplying or dividing by any fixed number to each element of a row.

Here I’ll solve an example for you.

#### Example of row transformation method to solve a set of equations

Disclaimer: This is not my own example. I have chosen it from a book. I have also given the due reference at the end of the post.

##### Example

According to Stroud and Booth (2011) “Given that

apply the method of row transformation to obtain the value of ”

##### Solution

Here the given set of equations is

First of all, I’ll write it in matrix form.

So it will be

Now I’ll introduce unit matrix as a coefficient of constant matrix

Thus it will be

So the combined coefficient matrix is

Now my first step is to move the unit matrix to the left-hand half of the coefficient matrix.

There is no fixed setup for that. One can use different ways.

###### Step 1

First of all, I’ll interchange row 1 and 2. This is because I want to bring 1 in the place of 3 in row 1.

So R1 R2 gives the coefficient matrix as

Now I’ll subtract thrice row 1 from row 2. Simultaneously I’ll also subtract twice row 2 from row 1.

Therefore R2 – 3R1, R3 – 2 R1 give the equivalent combined coefficient matrix as

Now I have already got

as a first column in the combined coefficient matrix.

My next step is to bring

as a second column in the combined coefficient matrix.

###### Step 2

Next, I’ll subtract row 2 from row 3.

Thus R3 – R2 gives the equivalent combined coefficient matrix as

Then I’ll subtract twice row 3 from row 2.

Thus R2 – 2 R3 gives the equivalent combined coefficient matrix as

Now I’ll add row 2 to row 1. Simultaneously, I’ll also subtract twice row 2 from row 3.

Therefore R1 + Row 2, Row 3 – 2 Row 2 give equivalent combined coefficient matrix as

Now I have already got

as a second column in the combined coefficient matrix.

My next step is to bring

as the third column in the combined coefficient matrix.

###### Step 3

First of all, I’ll add row 3 to row 1. Also, I’ll add row 3 to row 2.

So the combined coefficient matrix will be

Next, I’ll divide row 3 with 16.

Therefore the combined coefficient matrix will be

Now I’ll subtract 11 times row 3 from row 1. Also, I’ll subtract 8 times row 3 from row 1.

Thus the combined coefficient matrix will be

Now I have already got

as the third column in the combined coefficient matrix.

Now I’ll go to the last step of this problem.

###### Step 4

Now at the end I have

Therefore I can say

Hence I can conclude that is the solution set of this example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

## Leave a Reply