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## Linear algebra

### Course: Linear algebra>Unit 1

Lesson 6: Matrices for solving systems by elimination

# Using matrix row-echelon form in order to show a linear system has no solutions

And another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form. Created by Sal Khan.

## Want to join the conversation?

• I tried this problem on my TI-83 Plus and got a slightly different rref than Sal.
[ 1 2 0 3 0 ]
[ 0 0 1 -2 0 ]
[ 0 0 0 0 1 ]
I entered the coefficients as matrix A, and the constants as matrix B. Then, used the Augment function and saved it as matrix C. rref [C] gave me the above.

• The reason that your answer is different is that Sal did not actually finish putting the matrix in reduced row echelon form. For a matrix to be in RREF every leading (nonzero) coefficient must be 1. In the video, Sal leaves the leading coefficient (which happens to be to the right of the vertical line) as -4. Your calculator took the extra step of dividing the final row by -4, which doesn't change the zero entries and which makes the final entry 1.

Sal probably didn't put the final row in reduced form because he could see the contradiction coming. (0 = -4 is as much a contradiction as 0 = 1). He gets away with it in this case because he knows where the problem is going.
• Is the number of free variables equal to the number of dimensions for a possible solution? If you have 1 free variable, is the solution set a line? 2 free variables, a plane? Etc.
• Yes. You can write the solution space as a position vector plus each of the free variables multiplied by their own (Linearly independent) vectors, which gives you the span of those vectors. 1 free variable = span of 1 vector = line, 2 free variables = span of 2 vectors = plane, etc.
• can a matrix have multiple row reduced echelon forms?
• No, there's a theorem that says it's unique.
• the correction says 'The shapes described by the three orange equations are not parallel.'
why isn't there a solution then?
• Imagine 3 lines in 2d space, imagine they form a triangle. As you may guessed, the lines aren't parallel, and yet, they don't intersect, and by that I don't mean intersection of 2 lines(there are 3 points of that for a triangle), but intersection of 3 lines at some place, a point that lies on all of the three lines, that point doesn't exist in case of triangle.

It's not always like that, but I hope that this showed you that there might be systems of non-parallel equations with no solutions if the amount of equations is greater than 2.
• After watching all three reduced row echelon videos I don't understand the following things: what is an "augmented" matrix; why we can perform operations on the matrix without changing the solution; where reduced row echelon comes from (ie where it's form/rules come from); how you know if your solution is a plane, point, etc.; the significance of the term "pivot".
• Augmented forms of matrices have the "solution" (x+ y = n) IN it, usually represented as the last column, or an Ax1 matrix following the original matrix. Thus the "n", the last column representing the "solution" will change when performing row operations, but the value of the equations doesn't shave, as you're performing equal operations on both (all) sides of the =. Reduced Row Echelon Form just results form elementary row operations (ie, performing equivalent operations, that do not change overall value) until you have rows like "X +0Y = a & 0X + Y = b"
Concerning points, lines, planes, etc., this is generally only brought up for intuition's sake during early stages of matrix algebra, as it can get difficult to comprehend when you're working in higher dimensions. That being said, it depends on your solution and the dimensions you're working in what geometric representation is appropriate. In two dimensions, for example, constants (X, Y) represent a point, but in there dimensions it would be a line, with infinite points on the Z-plane (X, Y, aZ).
Pivot numbers are just the first non-zero entry in a row, significant to put matrices into the appropriate form (for example Reduced Row Echelon form).
• I still don't have the intuition of why pivot entries must be the first entry in a row. It could be a logical consequence of its definition but womehow I'm missing the point.

For instance, in the example Sal put forward at , why should the variable X1 (row 1, col.1) be the pivot entry and not X2 (row 1, col. 1)?
• If you get a 1 you are saying the coefficient on the variable is 1. So you know x1 = 3 instead of, say, 3*x1=9. For square matrices you get something like:
1 2 2 | 20
0 1 2 | 16
0 0 1 | 7
Then you know instantly that x3 = 7 and you can do back substitution easily to find the other variables. It also makes things easier for more advanced concepts in linear algebra.
• i don't understand it. because my english is not good.
• there are plenty of other sites that teach math. you can google the videos you desire. by setting google to your language you will recieve results in your native language first. :)
• at around Sal says "The pivot entry should be my only non-zero entry in my row". Does he mean column instead of row? If so, this is a repeating mistake throughout this video series
• Sal says that if a whole row is zero ([0, 0, 0, 0 | 0]), then the set of equations have an infinite amount of solutions. What is the intuition for that?

Is it because all the variables equal zero and therefor you can scale them indefinitely and get have an infinite amount of solutions?