We want to find inverse of the matrix

(1)

This means we want to find a matrix such that

(2)

We start reducing

(3)

multiply the first row by

(4)

Multiply the first row by and add to the second row, replace the second row

(5)

Next, multiply the second row by

(6)

Now add the second row to the first row and replace the first row

(7)

Now we see that

(8)

Let us check

(9)

Suppose we are given a linear system

(10)

This system can be written as

(11)

Or in the form where is the matrix of coefficients, is a variable column vector and

(12)

This is said to be a matrix equation. We can also write the given system in the form

(13)

We are in fact writing the vector as a linear combination of columns of . As you see, any matrix equation is consistent if and only if the vector can be written as a linear combination of columns of .

Suppose we are given the matrix equation in observation 1. If we find a matrix such that that is

(14)

then we could multiply both sides of the equation by and get

(15)

This way we are able to solve for . From example 1, we know

(16)

Therefore

(17)

Therefore and .

We want to solve the system

(18)

The associated matrix equation is

(19)

We know

(20)

Therefore, first we find

(21)

The first row has a leading so we move to the second row. Subtract first row from second, replace second

(22)

Next, multiply the second row with

(23)

Now we have a leading in the second row. Move on to the third row. Add first and third row, replace the third row

(24)

Multiply the second row by and add to the third row, replace the third row

(25)

Next, Add the third row to the second row and replace the second row

(26)

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

(27)

The only nonzero left other than diagonal element is the index . Subtract the second row from the first and replace the first one

(28)

Now we check to see if the obtained matrix is in fact the inverse of

(29)

Now we can compute

(30)

Therefore