5c. Rank Of a Matrix | GATE 1994 Question | Rank Nullity Theorem | Linear Algebra | Sachin Mittal

so let us now discuss about rank rank is one of the most important property that a metrics can have it is a single number it is a single number

that is very powerful which means that a matrix has one num one number which is called Rank and then this rank is one of the most powerful number that that

Matrix can carry which means this number or this rank can tell many things about a matrix so let's just see what is a rank so by definition rank is a number

of linearly dependent rows so this hash means number okay I mean this is just a short to write number so hash means number so this rank means that number of linearly independent rows or number of

linearly independent columns or number of pivot elements in the equivalent form or non-zero rows in equivalent form actually all of these are equal and by definition all of this is called rank

now uh like I will prove or I will give you the intuitive idea that why why these all are equal specifically why this is the number of rows and independent number of columns are equal I will give you the intuitive idea later

okay so this rank is defined like this now let's just quickly jump to the question so that you can understand what is this rank suppose if I ask you to find out the rank here as I told you this rank is

number of pivot elements in the equivalent form right in the equivalent form so ah number of the pivot elements in the equivalent form now let's just find out the number of pivot elements in the equivalent form so for that I need to First convert that to equilibrium

which means I need to have all the zeros here that are already already there all the zeros all the zeros all the zeros so okay uh this Matrix is already in the equivalent form now you just count the number of pivot elements so this is the

pivot element I think this is the pivot element this is a pivot element this is a pivot element so if this Matrix is M then I can say rank of this Matrix m is basically 4 why because there are four

pivot element that's how you can easily find out rank of this Matrix now let's solve this question they are asking the rank of a matrix and Matrix given is this this has been asking gate

1994. let's just solve this question

1994. let's just solve this question that how to find the rank so to find the rank you need to convert The Matrix into equivalent form and that procedure is called gaussian elimination right so uh let's just do it if that is 0 first 0

then can you make it non-zero by interchange yes you can interchange with the second or you can interchange with the third but we like to interchange with the third reason Wing is that we want to keep all the zeros at the last

so let me just say that R1 can be interchanged with R3 right and after this the operation after this operation you will be getting three one one

935 and 0 0 minus 3.

then you see that you need to make all of this 0 which means you can apply okay you can apply r r two should be replaced by R2 minus three r one I think that is

a good choice now if you do this you will be getting three one one which is as it is right okay let me just adjust this yeah you will be getting three one one

which is adjectives and then you will be getting this is 0 9 minus three or nine minus nine zero three minus 3 is also 0 and then 5 minus 3 will be 2 and then this

is 0 0 minus 3 right and then you need to make this also zero so I think you can easily make this zero and I think we have done already already this question now I remember right so you can definitely make this zero I don't care

about the operation but you can definitely replace the R3 by by something and by by making the zero so which means that ultimately you will be having two non-zero rules right this is

three one one this is zero zero two and then this is zero zero zero let's not worry about the operation that you will be having here now see here this is the pivot element this is the pivot element so there are two pivot elements that's

where the rank is two where is equation right so that's how easily you can find out the rank now whatever earlier examples we did like you can find the rank just using the question admission

so for example like we did some example here right so this was the example where we are converting this Matrix to equivalent form and if I if I just ask the rank of this Matrix then once you get the equivalent form let's just count the number of pivot element number of

pivot elements are two so that's why the rank of this particular Matrix is also two similarly if someone asks you rank of this particular Matrix okay we just did it this is also two right so that's how you can solve these questions for the rank you just need to convert to the

equivalent Matrix and count the number of pivot elements now rank is always positive it can be zero also okay it is like I can say non-negative so which means it would be positive or zero but

zero is only when the Matrix is zero Matrix which means if all are zeros then only the then only the rank is zero otherwise the rank is non-zero okay or n positive also because it is number of

pivot elements if the number of pivot elements are zero it means all must be zero then rank will be zero otherwise rank will never be zero so rank 0 means that that Matrix is zero Matrix or vice versa also okay I mean if Matrix is zero

Matrix then only rank would be zero or by server so if rank it zero it must mean then that Matrix is zero Matrix right now in the row equivalent form as I told you that rank of a matrix is equal to

number of pivot Elements which is also equal to number of linearly independent columns which is also equal to number of linear independent rules so this is just a summary that I mean this is just a same definition of the rank that we have

right now tell me this if I say number of pivot variables plus number of free variables are equal to total number of variables then I mean which means that the total number of variables you can rewrite it

number of pivot and three variables so for any Matrix you will be having either pivot variable I mean see for for any Matrix you will be having let's suppose this is corresponding to X this is corresponding to Y this is corresponding

to Z and let's suppose this is augmented column which is uh which is basically the vector B right this is this is augmented Vector B now for any Matrix you will be having either pivot variable

either X is a variable X is a private variable or axis of any variable suppose X is a private variable and Z is a private variable so either there are private variable and free variable so you can say that total number of variable which are three which are equal

to Pivot variable plus three variables which are also equal to total number of columns or rows what do you think so here it is always columns right because every column is corresponding to

variable so either I write variable or I write column both are same so every column is corresponding to variable so which means you you can say that total number of columns whatever you have okay let's suppose the hair you have three

columns so three can be divided into pivot columns and free columns which means period columns are two here three columns is one so for example if we have n columns then you can say there are some free

columns some pivot columns let's Suppose there are pivot columns then there will be n minus r free columns right and number of pivot columns is also called as rank number of pivot columns is also called as rank

I hope that is fine so what I mean to say here is that let me just tell you see what I'm saying that you have one Matrix where you have you have n columns right let's suppose

one two three let's suppose n columns I'm talking about the only coefficient Matrix this this is the extra column that we have now out of these n columns n columns you can say in coefficient Matrix okay in

coefficient Matrix which means not in the augmented Matrix obviously I never care about this as long as I am carrying over the pivot variable of the free variable because it

does not correspond to any variable okay the variables are here only x y z w or whatever the variable variables are here only so the total number of variables or the total number of columns can be divided into two parts free variable or

the free ah free column or the pivot variable or the pivot column let's suppose if the pivot column is R then free variable will be n minus r or free columns only n minus r so I will say

that this number pivot element on odd number of variables or number of columns are Rank and there is one more term which is called nullity nullity nullity is defined as number of free

variables okay so there is a term called nullity let me write it here nullity which is which is basically number of number of free variables

okay so let me write here rank is number of pivot variables number of good Columns of about variables or whatever you want to write and similarly

nullity is nullity is number of free okay number of free columns now suppose if this is R now suppose if rank is R then what about uh what about

nullity nullity if the total number of columns are n then what What will the nullity see the total number of columns will be divided between Rank and nullity I mean the pivot columns and the free

columns right rank anality if this is R then this will be n minus r or in other words if you just add both Rank and nullity you will be getting the total number of columns because because

if this is Pivot let's suppose let's suppose these are the pivot okay this is Pivot this is Pivot and and this is free okay let's suppose these two are free or maybe these three are free then you will say that these three will corresponding

to nullity these two will corresponding to rank and the total is five so that's why five columns are divided into rank anality right see that's very important to understand that you have n columns

you are you are saying out of those n columns some columns are free some columns are pivot and the pivot columns are called as Rank and the free columns are called anality so hence from here you can say that rank personality okay

sorry so from here you can say there is a theorem which says actually rank personality equal to n and what is this n here n is

total number of columns number of columns okay in a in a a means like a means coefficient

Matrix if I say a b it means augmented Matrix and a means just coefficient Matrix okay n a so this theorem is very nice now now this rank as I told you is number pivot

elements and this nullity is basically number of free elements or the free columns so this is free columns this is Pivot columns now this nullity is also defined as as some other thing which we don't need but

if you want I can just write the definition which is called dimension of the null space dimension of the null space actually this null space column space

and row space these things are not in the syllabus so we will not go to all of these details but your nationality is something which is called dimension of the null space basically there is a space which is R power n i mean this is

space r power n is divided into null space I mean some of the part is divided into null space and then column space okay anyway let's not worry about all of this maybe maybe that's not relevant but yeah

this this energy divided into rank a rank and led in the similar way you can say the spaces are divided into two parts I mean uh some some portion is called covered by analysis some portion is covered by column space anyway let's

not worry about all of this okay cool so here rank personality is n now I hope you understood this this sometimes useful theorem I mean if you add rank anality you will get total number of

columns and also that is obvious why because rank is Pivot columns analogies free columns now I hope you understood this ok so in the next video we will we will actually solve the system of linear

equations so till now we are just understanding few few terms here and there and then in the next video we will solve the system linear equations okay so thank you so much