Proof of elementary row operation for matrices?

[hitmankratos]

Hey everyone,

 

I'm taking a Linear Algebra course, and we just started talking about matrices. So we were introduced to the elementary row operations for matrices which say that we can do the following:

 

1. Interchange two rows.

2. Multiply a row with a nonzero number.

3. Add a row to another one multiplied by a number.

 

Now I understood from the lecture in class how to use these and all, but I want to understand the logic behind number 2 and 3?

Is there a mathematical proof that shows that by adding row R1 to row R2 we are not changing the system of equation? Same thing with number 2, how can we just multiply a row without changing the solution set?

 

 

Thanks in advance,

[DrRocket]

Hey everyone,

 

I'm taking a Linear Algebra course, and we just started talking about matrices. So we were introduced to the elementary row operations for matrices which say that we can do the following:

 

1. Interchange two rows.

2. Multiply a row with a nonzero number.

3. Add a row to another one multiplied by a number.

 

Now I understood from the lecture in class how to use these and all, but I want to understand the logic behind number 2 and 3?

Is there a mathematical proof that shows that by adding row R1 to row R2 we are not changing the system of equation? Same thing with number 2, how can we just multiply a row without changing the solution set?

 

 

Thanks in advance,

 

 

It should be obvious that those operations are invertible, with an inverse of a type contained in your list of allowable operations, and that any solution of the original set of equations is a solution after any of the named operations are performed.