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## Linear algebra

### Course: Linear algebra>Unit 1

Lesson 6: Matrices for solving systems by elimination

# Solving linear systems with matrices

Sal solves a linear system with 3 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form. Created by Sal Khan.

## Want to join the conversation?

• This method (converting to reduced row echelon form) seems somewhat incoherent. I'm not understanding the pattern to doing this. How is this possible without unbalancing the equations? • I wonder if there is a specific method we choose what operations to perform in the matrix when we try to reduce it,like if there is a method for example "first subtract the 1st row from the 2nd,then the 2nd from the multiple of the 3rd by 2 ",etc.I guess probably not,but i had to ask to be sure i m not missing something. • Is there a video that introduces the reduced row echelon form ? • I see Khan is using " = " sign but in my book we aren't allowed to use it between matrices, we use " ~ " instead. Can anyone explain that? •  `a ~ b` usually refers to an equivalence relation between objects `a` and `b` in a set `X`. A binary relation `~` on a set `X` is said to be an equivalence relation if the following holds for all `a, b, c` in `X`:
(Reflexivity) `a ~ a`.
(Symmetry) `a ~ b` implies `b ~ a`.
(Transitivity) `a ~ b` and `b ~ c` implies `a ~ c`.

In the case of augmented matrices `A` and `B`, we may define `A ~ B` if and only if `A` and `B` are augmented matrices corresponding to systems of equations having the same solution set. In this case `~` clearly is an equivalence relation. Since `A` may be different from `B` (they may have the same solution set, but they need not be the same system), writing `A = B` is not strictly correct. In short: use `~`.
• I've progressed through the videos, and don't recall him covering augmentation at any point. What is augmentation exactly? What video does he cover it in? • An augmented matrix simply means that there's that division between the part you have to reduce and the last column, which is the "answer".
{1+2+3
{3+2+1
would be represented in a "normal" matrix like this:
[1 2 3]
[3 2 1]
while
{1+2+3=6
{3+2+1=6
would have an augmented matrix, with the line instead of the = sign.
[1 2 3 |6]
[3 2 1 |1]
• The bird sounded beautiful! • When he said 3 unknown and 3 equations, did he mean that the x,y,z are the unknowns and the augmented values are the equations?

Please I don't know for sure. I don't want to guess on a quiz. • Is the reduced row echelon form the same as the guass jordan elimination? • Woah, woah, that's exactly why I'm watching this to begin with! And I've watched all the previous lessons, read the articles, and answered their questions successfully. And yet ... you lost me there, Sal. First of all, that's not how an "augmented matrix" looked like just a few articles/lessons ago; it was just a 2x3 matrix. Secondly, what on earth is reduced row-echelon form? And why does it seem in this video like augmented matrix = reduced row-echelon form?

This lesson feels like a jump and it's confusing me as someone who wants to understand what things are and why we're doing them. I understand that we're doing this to solve for x, y, and z, but pretty much everything else in this video is confusing now.

P.S. I came here via the "Introduction to matrices" series of videos at: https://www.khanacademy.org/math/precalculus/precalc-matrices  