Furry teaches you the Gauss-Jordan Elimination Method & Reduced Row Echelon Form

追加: 2026-05-05

[00:00:00]Hello everyone, MathCat here.

[00:00:03]Today, we will be discussing the Gauss-Jordan elimination method to then get to reduced row echelon form.

[00:00:12]If you're focused on purely computational methods, then feel free to skip ahead to the chapters listed below.

[00:00:19]All right. So, let's take a look at these two matrices.

[00:00:25]Do you remember um what kind of matrix this matrix is in?

[00:00:32]Take a moment to figure that out.

[00:00:36]This is called an augmented matrix.

[00:00:39]Notice that there's a line here. If we convert this back into its original form, we get 3x + 5y + z is equal to 3.

[00:00:53]And then so on and so forth.

[00:00:56]2z equal to 2.

[00:00:59]And then z is equal to 4.

[00:01:03]So, if you don't remember the conditions for row echelon form, there just needs to be a non-zero leading pivot in every row, and it needs to form a staircase, and there needs to be zeros underneath every single pivot.

[00:01:19]Now, is there anything different that you notice about this augmented matrix?

[00:01:26]This is in reduced row echelon form.

[00:01:31]Let's identify the criteria that qualifies or that lets a matrix be in reduced row echelon form.

[00:01:41]So, first criteria, every leading entry Let's Let's write that down.

[00:01:50]Every or the leading entry in each non-zero is one.

[00:02:12]So, if you look at every pivot or the non-zero leading number in every row, you'll notice that it's one.

[00:02:23]It'll never be in like this can't be three.

[00:02:27]This can't be 10.

[00:02:30]Each pivot has to be one in each non-zero row.

[00:02:35]Um okay?

[00:02:37]And um for reduced row echelon form, it follows the same principles as row echelon form.

[00:02:46]And in this case, they need to be like a staircase.

[00:02:51]So, each pivot is to the right of the pivot above it.

[00:02:56]This is to the right of the pivot above it.

[00:03:00]So, it forms a staircase pattern.

[00:03:03]And now here's the important distinction.

[00:03:07]Yeah, we should probably write that down.

[00:03:10]Each pivot is to the right of the pivot above it.

[00:03:31]Okay?

[00:03:33]So, um notice how there is a lot of zeros here.

[00:03:39]Um you'll learn later that this is something called the identity matrix, but let's just focus on reduced row echelon form for now.

[00:03:48]Notice that there are zeros under um each pivot, similar to row echelon form.

[00:03:56]There are zeros underneath each pivot, but there are also zeros above each pivot.

[00:04:07]So, zeros above and below.

[00:04:11]All right?

[00:04:14]And um let's write that down.

[00:04:26]So, a more precise way to define it is that each pivot is the only non-zero entry in its column.

[00:04:35]So, the only non-zero entry.

[00:04:54]Okay?

[00:04:57]So, um these are some of the main criteria that define reduced row echelon form. There's also one more criteria.

[00:05:08]If this matrix had like zero an entire row of zeros, then it adds another criteria to reduced row echelon form. If there is ever a row of zeros anywhere, like let's say the row of zeros was this row.

[00:05:33]Any all-zero rows need to be at the bottom.

[00:05:38]So, if you remember, you can manipulate systems of equations. You can move them around, do whatever you want to them.

[00:05:45]Um same thing applies here. Any all-zero rows need to be at the bottom.

[00:05:56]Does that make sense?

[00:05:58]All right. Let's go on to identifying some um matrices in the form of row echelon form or reduced row echelon form. Let's try and identify.

[00:06:13]Let's say you have a matrix one two one zero zero.

[00:06:33]Okay, let's say you have this matrix.

[00:06:36]Would you classify this as reduced row echelon form or something else?

[00:06:43]Well, if you asked me, I would identify this as reduced row echelon form.

[00:06:49]Why?

[00:06:50]Because if we take a look at each pivot column or every column that has a pivot, there are zeros.

[00:07:01]It's the only um non-zero component in its column.

[00:07:06]Notice that this row or this column doesn't have a pivot.

[00:07:10]Um this one has a pivot.

[00:07:14]This one has a pivot.

[00:07:16]And each pivot is to the right of the pivot above it. This pivot is to the right of this one.

[00:07:25]So, it forms a staircase pattern.

[00:07:29]And um if there were any all-zero rows, well, there's no all-zero rows, so we can ignore that criteria.

[00:07:42]Um but then we also see that every pivot is one.

[00:07:52]So, the leading entry in each non-zero row is one.

[00:08:00]All right.

[00:08:02]And you might be wondering, what is this? What is this two doing here?

[00:08:07]This looks weird.

[00:08:09]This is something called a free variable.

[00:08:13]So, that means that if we write out the solution for this, x plus 2y is equal to 2.

[00:08:27]Um let's do z is equal to 1.

[00:08:32]And well, let's do w.

[00:08:38]And then let's do z is equal to 3.

[00:08:41]So, I'm defining this as w, this as z.

[00:08:47]Notice how y doesn't have a unique solution.

[00:08:52]That means that a fixed value of x, because x is a pivot, plus y Y could be anything.

[00:09:04]So, y is equal to y.

[00:09:09]And these are the Let's rewrite this.

[00:09:16]x is equal to 2 - 2y.

[00:09:21]These will be the solution for this matrix.

[00:09:31]So, can identify that this is a free variable. These have unique solutions.

[00:09:38]This has a unique solution.

[00:09:40]But again, y could be anything. Oh, I didn't mean to circle this earlier. Um sorry about that. But um these have unique solutions, but y Y can be anything.

[00:09:55]That is basically what a free variable means. So, maybe geometrically um each every free variable adds one whole dimension of solutions. So, maybe for this there we have one free variable.

[00:10:12]That means um Y could be anything. Y could be any value. So, it adds an entire line of solutions. And let's say if you had another free variable that would turn that line into a plane.

[00:10:28]If there was two um free variables that make a plane of solutions that would work. There's not a specific like um that specific values that make the equation true.

[00:10:45]Any value that is defined by this line makes the equation true.

[00:10:54]So, instead of just dots that work for the equation, it's an entire line, entire plane if you add have two free variables or maybe three free variables, you have a full cube of solutions.

[00:11:11]So, a quick rule to remember this is the number of free variables is equal to number of unknowns minus number of pivots.

[00:11:40]So, um if you look here we have one free variable.

[00:11:46]But, let's say we didn't know that. We can make a generalization.

[00:11:51]We have 1 2 3 4 unknowns, X Y W Z.

[00:12:00]And we have only three pivots.

[00:12:03]So, we can for sure say that there will be one free variable because four unknowns minus one pivot, oh sorry, minus three pivots.

[00:12:20]This is equal to one free variable.

[00:12:23]Let's talk about possible cases for reduced row echelon form and what those cases mean.

[00:12:33]So, there are three cases.

[00:12:37]Let's say you have this matrix.

[00:12:48]This is in reduced row echelon form and we can see here that there's a row of zeros. If you ever see a row of zeros, you want to put it at the bottom and that would mean that this has infinite solutions.

[00:13:13]Because zero is equal to zero. If you write this whole thing out X + Y equal to one and zero plus zero is equal to zero.

[00:13:25]Zero equals zero.

[00:13:26]Infinite solutions. Okay?

[00:13:31]Now, let's take a look at second case.

[00:13:44]You will see that if you write this out X is equal to three.

[00:13:51]Y is equal to one because one X is equal to three.

[00:13:57]Y or one Y is equal to one.

[00:14:00]You'll see that each variable has a unique solution. So, this matrix has one unique solution.

[00:14:16]Now, let's take a look at the last example.

[00:14:48]So, if you write this out, you'll immediately see that there's a contradiction.

[00:14:55]This Z here saying it's equal to zero. Actually, no, this should be switched. I'm so sorry.

[00:15:07]You'll see that it says like zero is zero X plus zero Y plus zero Z is equal to one.

[00:15:14]That's impossible. So, there's a contradiction. It doesn't make sense.

[00:15:19]Therefore, there's no solution.

[00:15:24]Okay?

[00:15:27]Now, um here is a case that I would like to cover.

[00:15:32]If you ever get something like this you'll see that there's a zero zero zero row that usually means that there's infinite solutions, right? But, because there's a contradiction up here that immediately overrides that infinite solutions. So, this matrix has no solution.

[00:16:02]The contradiction will always override any matrix causing that to be no solution.

[00:16:10]Is this in reduced row echelon form?

[00:16:15]Well, I would say that yes, this is in reduced row echelon form because there is um non-zero pivots in each column.

[00:16:30]So, for this one um but there is also an entire row of zeros and it's at the bottom.

[00:16:39]Okay?

[00:16:40]And we can count the amount of free variables.

[00:16:44]Um again, let's do number of unknowns minus number of pivots.

[00:16:57]There are three unknowns.

[00:17:01]X Y Z.

[00:17:05]Um and there are two pivots.

[00:17:08]So, that means there is one free variable. So, could be Z.

[00:17:16]And you set that equal to itself and that is an entire line of solutions.

[00:17:22]Isn't that cool? Okay, let's move on.

[00:17:28]So, given a matrix we want to convert this to reduced row echelon form.

[00:17:47]First, let's convert it to row echelon form. You must convert a matrix into row echelon form before you actually start computing into reduced row echelon form.

[00:17:59]Okay?

[00:18:01]So, let's get rid of the second row here by Let's start by multiplying the bottom row by negative two.

[00:18:17]So, row one minus two row two is equal to the new row two.

[00:18:30]Two four eight.

[00:18:34]And then negative two times one is negative two.

[00:18:40]And that means two um minus two is zero. So, that becomes zero.

[00:18:49]And then three times negative two is negative six.

[00:18:56]Plus four is negative two.

[00:19:01]And then negative two times seven is negative 14.

[00:19:09]Negative 14 plus eight that should be What is that?

[00:19:33]Negative six. Okay.

[00:19:37]And notice how everything here is divisible by two.

[00:19:43]So, um this is equivalent to this.

[00:20:04]Now, how do we This is in row echelon form, but how do we actually reduce this into reduced row echelon form?

[00:20:17]Well, we want to get rid of this up here. We want to get this pivot down here to be the only non-zero entry.

[00:20:27]So, we can take row two, multiply it by two, and add it to row one two get the new row one.

[00:20:46]Okay?

[00:20:49]So, um two times zero, that is unchanging. Let's write bottom row first.

[00:21:04]Two times negative one is negative two plus two that is zero.

[00:21:12]Negative three times two negative six plus four is negative two.

[00:21:23]Right? I know.

[00:21:25]Negative two.

[00:21:27]So, now we can identify that this is X, this is Y.

[00:21:37]X is equal to negative two.

[00:21:41]And negative Y is equal to negative three, which is Y is equal to three.

[00:21:48]And this is your solution.

[00:21:51]Does this have infinite solution, a unique solution, or no solutions?

[00:21:57]Well, this is in the form of unique solutions.

[00:22:00]We just identified those.

[00:22:03]Now, let's practice with a three by three matrix. Okay?

[00:22:13]Actually, I like three by four matrix. Okay.

[00:22:19]Now, let's see. This is in already in row echelon form.

[00:22:27]Let's try and get this pivot to one.

[00:22:32]So, one half of row three is equal to the new row three.

[00:22:53]Now, this is we have the pivots as one.

[00:23:00]Now, we just want to get rid of these.

[00:23:03]So, let's start from row three.

[00:23:06]So, we can multiply row three by by what?

[00:23:19]By negative two.

[00:23:22]And then we add it to row two, and that gets us our new row two.

[00:23:31]Okay?

[00:23:35]One two one eight.

[00:23:40]And then uh zero times negative two zero.

[00:23:47]And then negative two times one um plus two is zero.

[00:24:00]And then negative two plus two or sorry, negative two times two is negative four plus seven is three.

[00:24:12]And then we want to get rid of this one up here.

[00:24:23]So, we can just do row three or minus row three plus row one.

[00:24:33]And that gets us that removes that one up here. So, then this pivot can be alone in this column.

[00:24:42]So, one two zero and then negative two the same is six.

[00:25:03]Now, we only need to get rid of one more up here.

[00:25:10]How do we do that?

[00:25:11]We utilize row two.

[00:25:17]Row two or two of those um negative two times row two plus row one is equal to the new row one.

[00:25:34]Zero zero cuz the negative two times one plus two is zero.

[00:25:43]And then everything is zero.

[00:25:46]So, then negative two times three which is negative six plus six that is zero.

[00:26:05]Now, we have our solution set.

[00:26:08]Does this have one solution, one unique solution, infinite solutions, or no solutions?

[00:26:16]I'd say it has one unique solution because every variable has its corresponding output, and there's no contradictions.

[00:26:29]Now, let's test your concepts.

[00:26:35]Let's say you have a matrix.

[00:26:43]Without row reducing, explain why the system must have infinite solutions.

[00:26:52]Well, you can see here if we add two of these, we do negative two row one plus row two to get the new row two, you will see that if you want to perform this operation, apply negative two to everything, you'll get this row of zeros, which you can then move to the bottom, and zero is equal to zero.

[00:27:33]This has infinite solutions.

[00:27:39]Let's test another concept.

[00:27:41]Let's say you have a three by five augmented matrix.

[00:27:47]It has three pivots.

[00:27:50]How many free variables are there?

[00:27:54]Well, we can use this formula or this Yeah, I guess this formula.

[00:28:03]The number of free variables is equal to the number of unknowns minus number of pivots.

[00:28:21]So, a three by five matrix has five unknowns because if you expand it out, um you have one, two, three rows and one, two, three, four, five um columns.

[00:28:41]Five unknowns.

[00:28:43]Actually, I'm sorry.

[00:28:45]It actually has four unknowns because this is an augmented matrix. I forgot to consider um this equals here.

[00:28:56]So, you actually have one, two, three, four unknowns and one output here. So, there are four unknowns and three pivots.

[00:29:10]That means there will be one free variable.

[00:29:20]Let's test one more concept.

[00:29:24]Let's say a system has more equations than unknowns.

[00:29:29]Does that guarantee a unique solution?

[00:29:33]Let's say you have something like this.

[00:29:46]Hm.

[00:29:51]>> Well, you'll see that there's a redundancy.

[00:29:56]There are two unknowns, but three equations.

[00:29:59]This is just a multiple of this.

[00:30:02]So, it doesn't exactly guarantee a unique solution.

[00:30:07]Cuz I mean, this could have a unique solution, but let's say you have something like this.

[00:30:18]There's a contradiction.

[00:30:20]And um that means there is no solution.

[00:30:26]Or let's say you had one two equal to 100.

[00:30:34]This and this, this is a contradiction.

[00:30:40]They don't equal each other.

[00:30:42]Doesn't make sense.

[00:30:44]Right?

[00:30:46]There's no solution.

[00:30:48]So, a system having more equations than unknowns does not guar- antee a unique solution.

[00:31:28]Nice. Hello, everyone.

[00:31:30]Thank you guys so much for all this fan art.

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[00:31:51]Differential equations is starting to ramp up as well as some of my other classes. So, I will be trying my best to finish the linear algebra series, but do know that uploads will slow down for now.

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