# Rank of the augmented matrix

#### diogomgf

##### Junior Member
$$\displaystyle A$$ is any matrix $$\displaystyle M_{m \times n}$$, and $$\displaystyle B$$ is a matrix based on any column of $$\displaystyle A$$.
How can I justify that their ranks are the same, $$\displaystyle r(A) = r([A|B])$$ ?

#### Romsek

##### Full Member
and $$\displaystyle B$$ is a matrix based on any column of $$\displaystyle A$$.
what does this mean?

#### diogomgf

##### Junior Member
what does this mean?
That $$\displaystyle B \in M_{m \times 1}$$, where $$\displaystyle B$$ is a random column of $$\displaystyle A$$. It is supposed to be irrelevant...

#### Jomo

##### Elite Member
Because B is not included in calculating r (A|B)

#### Jomo

##### Elite Member
In all cases?
I will let you decide. Was there anything special about your A and B or were they just arbitrary matrices?

By the way, how do you determine the rank of a matrix? How about the rank of a system of equations?

#### diogomgf

##### Junior Member
I will let you decide. Was there anything special about your A and B or were they just arbitrary matrices?

By the way, how do you determine the rank of a matrix? How about the rank of a system of equations?
Well my O.P was to understand why in this specific scenario B doesn't influence the outcome of the rank of the system of equations.

To determine the rank you just transform the matrix A (as in [A|B]) to row echelon form and see the number of non-null rows obtained.
The same can be said about the augmented matrix. Sometimes the matrix B obtained does influence the rank of the matrices (I asked "in all cases?" but I shouldn't have )...

#### Jomo

##### Elite Member
How does the matrix B obtained does influence the rank of the matrices?

#### diogomgf

##### Junior Member
How does the matrix B obtained does influence the rank of the matrices?
Thats what I'm trying to know

#### Jomo

##### Elite Member
Thats what I'm trying to know
How do you determine the rank of a matrix? How do you determine the rank of a system of equations? You answered those--thank you.
Now I have a different question. What is the differences, if any between the two methods?

#### diogomgf

##### Junior Member
How do you determine the rank of a matrix? How do you determine the rank of a system of equations? You answered those--thank you.
Now I have a different question. What is the differences, if any between the two methods?
There isn't any difference. You just have to put the system of equations as an augmented matrix and find the rank...

#### Jomo

##### Elite Member
There isn't any difference. You just have to put the system of equations as an augmented matrix and find the rank...
So use that as your justification as why the two ranks are equal?

#### Otis

##### Senior Member
So use that as your justification as why the two ranks are equal?
I don't understand that question.

#### diogomgf

##### Junior Member
So use that as your justification as why the two ranks are equal?
Got it ! Thank you