Rank of the augmented matrix

diogomgf

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\(\displaystyle A\) 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

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diogomgf

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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

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

Jomo

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In all cases? :confused:
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

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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 :LOL:)...
 

Jomo

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How does the matrix B obtained does influence the rank of the matrices?
 

diogomgf

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Jomo

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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

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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

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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

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