Need up to 30 seconds to load.

the rank of a matrix is the number of

linearly independent rows that it has or

equivalently the number of linearly

independent columns the rank of a matrix

is always positive except in the case of

the zero Matrix which has rank 0. let's

go through five examples of calculating

the rank of a matrix we'll do the first

four real quick and we'll be a bit more

thorough with the last one here's a

matrix a it's not the zero Matrix so its

rank is positive generally to find the

rank of a matrix we'll want to put it

into row Echelon form doing this to a

gives us this Matrix and this tells us

the number of linearly independent rows

that a has it's simply the number of

non-zero rows in the row Echelon form in

this case we see that's three so the

rank of a is equal to three similarly

for this Matrix B if we put it into row

Echelon form this is what we get the

number of linearly independent it rows

in B is the number of non-zero rows in

the row Echelon form in this case there

are two non-zero rows so the rank of B

is two some matrices are simple enough

that we don't need to put them in row

Echelon form this Matrix C is not the

zero Matrix so it does have a positive

rank but I notice the two rows are not

independent the second row is simply

equal to negative two times the first

row so C has only one independent row

and its rank which we could also denote

like this is equal to one this Matrix D

has fewer columns than it has rows so it

may be simpler to look at the columns we

notice the first two columns are

linearly independent there's no way to

multiply column one to get column two

however column three is a linear

combination of the first two columns in

fact if we just add column one in in

column 2 that gives us column three so

the Matrix D has only two linearly

independent columns and we would say

that the rank of D is equal to two if

all else fails we can always put the

Matrix into row Echelon form and count

the non-zero rows let's do one more

example of that with this Matrix e we

want to make the entries below each

leading entry equal to zero so below

this two we want only zeros to get that

we'll set row two equal to row two minus

row four so those ones will cancel out

to get rid of this one we will subtract

row two from row four to again cancel

out the ones that gets us here we have

zeros below the two like we want now we

want to get zeros below the next leading

entry of negative one to make this two

equal to zero We'll add two copies of

row two into this third row and that

gets us here now we also want to get rid

of this one that's below the leading

entry of negative one and in order to do

that we'll simply add Row 2 to Row 4

which will eliminate Row 4 completely so

Row 4 is all zeros and we can see we are

now in row Echelon form and there are

three non-zero rows thus the rank of E

equals three thanks for watching let me

know if you have any questions