Triangular decomposition method in 3 × 3 matrices. Hello friends, here it’s all about the triangular decomposition method in 3 × 3 matrices. Have a look!

### Triangular decomposition method in 3 × 3 matrices

There are several ways to solve a set of equations in matrix algebra like the Gaussian elimination method, row transformation method, inverse matrix method and so on. The triangular decomposition method is also one of those.

See also: Gaussian elimination method in 3 × 3 matrices

Gaussian elimination method in 4 × 4 matrices

Here I’ll only work with the 3 × 3 matrices. Soon, I’ll also write on the triangular decomposition method in 4 × 4 matrices.

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

##### Method

Suppose I have a set of equations like

Now I have to solve these equations using the triangular decomposition method.

###### Step 1

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

Thus it will be

So I can say the system of equations is in the form of .

Here is the coefficient matrix, is the variable matrix and is the constant matrix.

In the method of triangular decomposition, the first task is to write the coefficient matrix as a product of two matrices and .

The matrix will be a lower triangular matrix. And, the matrix will be an upper triangular matrix.

Thus it will be .

Here the matrix will look like

Similarly, the matrix will be

Thus the system of equations will be

###### Step 2

Now I’ll choose .

Here the matrix will be

So I can say

Then I’ll solve for . Next, I’ll solve for using the relation .

Now I’ll solve an example of that.

#### Example of the triangular decomposition method in 3 × 3 matrices

Here is the example of the triangular decomposition method in 3 × 3 matrices.

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

##### Example

According to Stroud and Booth (2011) “Using the method of triangular decomposition, solve the following set of equations.

”

##### Solution

Here I know the set of equations as

So I can say that the coefficient matrix is

Now my first job is to write the matrix as a product of two matrices and .

So I choose the lower triangular matrix as

Next, I choose the upper triangular matrix as

Therefore it will be .

So that means

Now I’ll get the matrices and by solving this system.

###### Step 1

I’ll start with matrix multiplication.

First of all, I’ll multiply both the matrices and to get

So now I have nine equations to solve.

I’ll start with the first row.

###### Step 2

The first one is the element from the first row and the first column.

(1)

Next, is the element from the first row and the second column.

So it will be

From equation (1), I can already say .

(2)

Now comes the element from the first row and the third column.

So it will be

From equation (1), I can already say .

(3)

Now it’s time for the second row.

###### Step 3

Next, comes the element from the second row and the first column.

(4)

Now comes the element from the second row and the second column.

So it will be

From equations (2) and (4), I can already say .

Thus the value of will be

This means

(5)

Next, comes the element from the second row and the third column.

So it will be

From equations (3), (4) and (5), I can already say .

Thus the value of will be

This means

(6)

Now comes the third row.

###### Step 4

First comes the element from the third row and the first column.

(7)

Next comes the element from the third row and the second column.

So it will be

From equations (2) and (7), I can already say .

Thus the value of will be

This means

(8)

At the end comes the element from the third row and the third column.

So it will be

From equations (3), (6), (7) and (8) I can already say .

Thus the value of will be

This means

(9)

Now I’ll use equations (1) – (9) to write the matrices and .

Thus it will be

and

###### Step 5

Now I’ll choose .

Here the matrix will be

So I can say

Now I’ll solve for .

Thus it will be

Next I’ll do the matrix multiplication on the left-hand-side.

Therefore it will look like

So I get three equations now.

(10)

Next equation is

Using equation (10), I can say that

(11)

The last equation is

Using equations (10) and (11), I can say that

(12)

So I can say

###### Step 6

Now I’ll solve for using the relation .

From Step 4, I already know that the matrix is

Thus it will be

Again I’ll use matrix multiplication on the left-hand-side.

Therefore it will be

Like Step 5, here again I get three equations like

From the last equation, I already know that .

From the middle equation, I can say that .

This gives

The first equation gives

This means

Hence I can conclude that by using the triangular decomposition method in 3 × 3 matrices I get the solution of the set of equations as

This is the answer to the given example.

Dear friends, this is the end of 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!!

Maarten MOSTERT says

Hi,

I am programming B-spline and C-splines and your blog helped me to find the missing bits and pieces to make them all work.

Good work.

Maarten,

Dr. Aspriha Peters says

Hi Marteen,

Thank you very much. It means a lot to me.