Suppose we want to find a sequence of elementary matrices that can be used to write

(1)

in row-echelon form. We know an by matrix is elementary if it can be obtained by applying exactly one elementary row operation to the by identity matrix that is interchanging two rows, multiplying a row by constant or multiplying a row by a constant and adding to another row. Next we can multiply the matrix on the left with each elementary matrix to apply the desired row operation to . To start we want to interchange row and so take

(2)

and write

(3)

Next, we want to have the leading in the first row, so take

(4)

and write

(5)

and therefore we have in row-echelon form.

Suppose we want to find inverse of the matrix

(6)

using elementary matrices. We start by writing each row operation. First, we have a leading in the first row so add first and second row and replace the second row

(7)

The elementary matrix associated to this step is

(8)

Next, multiply the second row by and add to the first row

(9)

The elementary matrix associated to this step is

(10)

Now we see that

(11)

Therefore

(12)

that is

(13)

Now we check

(14)

Suppose, if possible, we want to write the matrix

(15)

as a multiplication of a lower triangle matrix and upper triangle matrix that is Start by using elementary matrices to put in row-echelon form

(16)

(17)

Next,

(18)

(19)

Next,

(20)

(21)

Next,

(22)

(23)

Now, observe that each elementary matrix is invertible and we can write

(24)

Now we get

(25)

Now we check the factorization

(26)

Suppose we are given the system

(27)

and we would like to use -factorization to solve this system. First note that we can write this system as

(28)

that is

(29)

where

(30)

Please observe that when we can write

(31)

then

(32)

This gives us a hint. Define

(33)

and so solve the system

(34)

that is

(35)

After this,

(36)

can be solved for . Let’s try it. From example 3, we know

(37)

We write

(38)

Now solve

(39)

This provides

(40)