Question 1

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?

Accepted Answer

Basically, you're trying to find one pivotal entry, which must be in their own columns from left to right, while reducing the other coefficients to 0. You can use basic arithmetic to reduce the numbers in the matrix, by performing matrix row operations.

Question 2

Is there a video that introduces the reduced row echelon form ?

Accepted Answer

These videos seem to be split between the playlist in Math/Precalculus/Matrices and the playlist in the Math/Linear-Algebra/Vectors-and-Spaces unit https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces. 
You'll find the videos on row echelon form under the section "Matrices for solving systems by elimination", and specifically, the video which is supposed to go before this one is here: https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/matrices-elimination/v/matrices-reduced-row-echelon-form-1

Question 3

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.

Accepted Answer

There is no specific method to simplify any system of three equations. One simply has to look at the equations that one is given and determine how to make the X, Y, and Z zero based on that system's coefficients. 
Fascinating question.

Question 4

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?

Accepted Answer

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]

Question 5

1:11 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?

Accepted Answer

`*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 `~`.

Question 6

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.

Thanks for your time!

Accepted Answer

You are correct: x, y, and z are the unknowns. The augmented matrix contains the coefficients of the unknowns on the left.

Question 7

Is the reduced row echelon form the same as the guass jordan elimination?

Accepted Answer

Reduced row echelon form is what Gaussian Elimination achieves. So Gaussian Elimination is the method, reduced row echelon is just the final result.

Question 8

Sorry I'm not here to challenge anyone, but just confused. Because, what's the point of converting to row-echelon form? Using the equations in the video, x+y+z=3 can be rewritten as  x=3-y-z. And so we can substitute x=3-y-z into equation 2, then we can find out that y=-3-2z. Finally substitute x=3-y-z and y=-3-2z into the 3rd equation, then we can find out the value of z. After finding out the value of z, we can find out everything else by substituting the value of z into other equations. So why would we use the row-echelon form instead? Is finding out the values of x, y and z the only purpose?

Accepted Answer

Because substituting like that requires a little creativity, and we can often find shortcuts if the coefficients happen to look right. Our goal with row-echelon form is to develop a purely mechanical way to solve linear systems of equations that we could potentially give to a computer to execute, and that works with any coefficients we may have.

Algebra (all content)

Course: Algebra (all content) > Unit 20

Solving linear systems with matrices

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