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This is a video on how to find the rank of a matrix just real real quick and easy just for you know if youre first starting out with youre not really sure whats going on so lets look here at this first matrix here. A and lets before we go on before we even look at the ring. Lets look at what the measurements of this is this has 1 2.
3. Rows and 1 2. 3.
Columns. So that tells me its measurement is 3 by 3 and this is either usually noted as n by n or n by m use n by n. If its square square means.
It has the same dimensions as you can see and then n by m. If its not square n. By m.
Would be like this is 1 2. 3. By.
6. So this would be n by m. Over here.
Which would be 3 by 6. This would be 3 by 3 and one thing i want to point on about the rank is that the rank the maximum the rank will be is the number of rows that have been given or the number of columns depending on what order youre in so. Its either gonna be the maximum of this or the maximum of this and youll see here in a second.
Why that is and lets look here well.
If you look at matrix. A here we have this one here. And this one.
Here these are called pivot points. And the total number of pivot points. Is equal to the rank.
So this has two pivot points. One here and one here so this is rank equal to 2. Now what about this here.
Weve got 3 rows and 6 columns well as you can see if i were to reduce this down and get it into this form the maximum the maximum rank that this could have would be 3 because we would have 0 0 0 0 0 0 everywhere here just like that so the maximum rank this could be would be 3. Which is the number of rows. But if we did whats called transpose.
If we transpose with this thing. This would be equal to 1 1. 1.
And then zeros and zeros again. Where this would be 3. This way and then 6 this way so we just transposed.
It that flips this it becomes a 6 by 3. Now and still the maximum number of rank would be 3. So you can see that whatever this smaller number is and your dimensions is going to be the maximum number of the rank.
But we asked ourselves what is the rank of b.
Well. I tell you right now the rank of b. The rank of b is equal to 2.
Although you say hey how is that equal to 2. When youve got these three here well. This is not this this reduced down.
Actually kicks out this one here and so its 1 1. The reason why i know that is well if you started doing row. You started doing row operations and matrices you can see here that this row here is twice this row.
Here. So. This is 2.
This is 4 3. 6. 4.
8. And so on and you see this has a negative value here this has a positive these are all negative. These are all positive well if you do this were gonna say b and were gonna perform row 3 plus row 1 is equal to row 3.
So i this is part of row operations. You can add rows you can subtract rows. So this tells me row 3 plus row 1 is equal 3.
This is gonna be replaced so if i do that im gonna say im actually gonna do 2 times row 1.
So if i multiply. 2 by this whole row. Im gonna get 4 6.
8. 10. 12 and negative.
14. And that added with this row is all zeros and so then i would replace that right here we would have all zeros. There and id be left with 1 2.
And that would be my answer for that would be ranked. 2. Now hopefully take a second and look at this one over here.
And you can see that this is going to be ranked 1 for the same reason. This is ranked 2. So the rank of h is equal to 1 reason.
Why is because these are all multiples of each other so if i if i do a row. If i do 4 h. I do row 1 minus row.
2. And then i do row 1 minus. Row.
And i say that becomes the new row. 2. And this becomes the new row.
3. Then im left with h equaling two two two and then 0 0 0 0. 0.
And as you can see if i divide this by 1 or by 2. Then this becomes 1 1. 1.
Those are all zeros and then youre left with rank. 1. Heres your 1 pivot point.
Remember the pivot point has to have a has to have 0 below it what in a if it goes to the next row. It has to have a zero to the left of it and then a zero below it and and so on so if we lets look at a matrix. That is called a full rank so a full rank lets lets say.
This is a five by five. So thats gonna be one two three four five thats full rank and these are all zeros and all zeros. So you can see theres a zero to the left of it that tells you that its that its a rank.
And theres a zero below. It so if theres a zero to the left and zero below. It then you know with the exception of the first one the first pivot.
Then thats your ring. So this would be rank five for this these are all zeros here. So thats how you would just kind of look at it analytically.
You know analyze it or by inspection. .
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