Today it’s all about Gaussian elimination method in 4 × 4 matrices.

### Gaussian elimination method in 4 × 4 matrices

#### Example of Gaussian elimination method in 4 × 4 matrices

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

So here is the example.

##### Example 1

According to Stroud and Booth (2011)* “Solve the following set of equations by Gaussian elimination:

##### Solution

Here the given set of equations is

I can rewrite it in matrix form as

Thus the coefficient matrix is

Also, the constant matrix is

Therefore, the augmented matrix will be

Like my earlier post on Gaussian elimination method in matrices, here also I’ll start with the augmented matrix.

Also check it out: Gaussian elimination method in 3 × 3 matrices

My first task is to reduce the augmented matrix to an upper triangular matrix.

For that, I’ll try to get three zero in the first column of the augmented matrix.

###### Step 1

Now the augmented matrix is

First of all, I’ll subtract thrice row 1 from row 2.

Simultaneously, I’ll also subtract 4 times row 1 from row 3.

Together with that, I’ll subtract row 1 from row 4.

In mathematical term, I’ll write it like this:

Row 2 – 3(Row 1), Row 3 – 4(Row 1), Row 4 – Row 1.

Thus the equivalent matrix will be

Now I have already got three zeros in the first column.

Next, I’ll try to get two zeros in the second column where I already have zeros in the first column.

###### Step 2

Therefore, as a next step, I’ll subtract twice row 2 from row 3.

Also, I’ll subtract th times row 3 from row 4.

In mathematical term, it will be

Row 2 – 2(Row 3), Row 4 – (Row 3).

Thus the equivalent matrix will be

Now I’ll divide row 3 with .

Therefore the equivalent matrix will be

Next I’ll interchange between row 2 and 3.

Therefore the equivalent matrix will be

Now I have got two zeros in the second column where I already had zeros in the first column.

My next task is to get one zero in the third column where the elements at both first and second columns are zero.

###### Step 3

Therefore, as a next step, I’ll subtract th times row 4 from row 3.

In mathematical term, it will be Row 3 – Row 4.

Therefore the equivalent matrix will be

Next I’ll interchange between row 3 and 4.

Therefore the equivalent matrix will be

Thus finally I have got an upper triangular matrix.

Therefore the system of equations in the matrix form is

Now my next job is to solve this system.

###### Step 4

Here I’ll use the backward substitution to solve the system of equations.

Now the equations are

(1)

(2)

(3)

(4)

Therefore from equation (4), I can say that

Next I’ll substitute in equation (3) to get

Now I’ll simplify it to get the value of .

I’ll put and in equation (2) to get

Now I’ll simplify it to get the value of

Thus it will be

At the end, I’ll substitute and in equation (1) to get

Hence I can conclude that and is the solution of the given set of equations.

This is the answer to 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!!

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