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in this video we're going to talk about

how to determine the inverse of a 3x3

matrix so let's say if we have matrix a

and it contains the elements 1 2

negative 1 negative 2 0 1 1 negative 1

and 0 so go ahead and determine the

inverse of matrix a so first we need to

rewrite this in the form of an Augmented

matrix with matrix a and the

multiplicative identity of a 3x3 matrix

so is matrix a it's 1 2 negative 1 and

so forth and the multiplicative identity

matrix it's going to be 1 0 0 0 1 0 0 0

1 so now here's what we need to do we

need to perform elementary row

operations and basically take the left

side and make it look like the right

side now the operations that we apply to

the left side we have to apply to the

right side so the right side is going to

change as a result and whatever the

result is on the right side that is

going to be the inverse of matrix a so

let's begin by turning this number into

a 0 to do that let's subtract Row 1 by

Row 3 and let's apply the result to Row

3

now Row 1 is not gonna change so let's

rewrite that Row 2 is also not going to

change as well now for Row 3 it's gonna

be 1 minus 1 so that's 0 and then it's 2

minus negative 1 or 2 plus 1 which is

stream and then we're gonna have

negative 1 minus 0 so that's negative 1

and then 1 minus 0 is 1 0 minus 0 is 0

and then 0 minus 1 is negative 1 so now

what do you think we need to do next how

can we turn this negative 2 into a 0

what row operations do we need to apply

to row 2 to do so so we're going to say

2 times R 1 plus R 2 let's do that so

we're change in Row 2 Row 1 is going to

stay the same Row 3 will remain the same

as well so 2 times 1 plus negative 2

that's going to be 0 2 times 2 plus 0

that's going to be 4 2 times negative 1

which is negative 2 plus 1 that's

negative 1/2 times 1 plus 0 is going to

be 2 2 times 0 plus 1 that's 1 2 times 0

plus 0 is 0 now let's turn that number

into a 0 so we need to apply the row

operation to Row 3 I'm gonna multiply

our 2 by 3 because 3 times 4 is 12 and

then I'm gonna subtract it by 4 times R

3 because negative 4 times 3 is negative

12 nope

so it's a zero so Row one it's gonna

stay the same and let's rewrite Row two

so we're gonna have three times zero

minus 4 times 0 which is 0 and then 3

times 4 minus 4 times 3 that's going to

be 0 as well and then 3 times negative 1

which is negative 3 minus 4 times

negative 1 so we have negative 3 plus 4

that's gonna equal to 1 and then 3 times

2 is 6 minus 4 times 1 so 6 minus 4 is 2

3 times 1 minus 4 times 0 that's 3 & 3

times 0 minus 4 times negative 1 is 4

now what we need to do is to turn this

number into a zero so that's going to be

straightforward we just got to add in

Row 1 and Row 3 to accomplish that

so let's rewrite rows 2 & 3

so it's gonna be 1 plus 0 which is 1 and

then 2 plus 0 that's 2 negative 1 plus 1

is 0 and then 1 plus 2 is 3 and then 0

plus 3 is 3 and 0 plus 4 is 4 now let's

focus on that number what row operations

do we need to apply to Row 2 in order to

make it 0 all we need to do here now is

add Row 2 and Row 3 so notice what's

going to happen so let's rewrite Row 1

and let's rewrite row three

so Row 2 plus Row 3 0 plus 0 is 0 and

then 4 plus 0 is 4 negative 1 plus 1

that's going to be 0 and then we have 2

plus 2 which is 4 and 1 plus 3 that's 4

and then 0 plus 4 is also 4 now we'll

need to do at this point

wait hold on I didn't copy this

correctly

this is supposed to be 1 2 0 & 3 3 4

so everything else should be fine at

this point this is what I should have

right now so at this point we need to

turn this number into a zero so we need

to apply the operation to Row one so

let's take two times R one and subtract

it by our two so we're going to have

zero zero one two three four and then

zero four zero four four four now we're

going to take two times one minus zero

so that's gonna stay one and then two

times two minus 4 that is now 0 2 times

0 minus 0 is 0 and then 2 times 3 which

is 6 minus 4 that's 2 the next one's

gonna be the same 2 times 4 is 8 minus 4

is 4

and one thing I do need to fix two times

one minus zero that's supposed to be two

and have a 1 here so let's fix that now

we need to do is multiply the first row

by 1/2 and the second row by 1 over 4

and that should give us what we need so

half of 2 is 1 and half of 4 is 2 1/4 of

4 is 1 so all of these will be 1 and

then the last one it's not going to

change so notice that here we have the

multiplicative identity matrix I 3 and

this side represents the inverse of

matrix a 1 1 2 1 1 1 2 3 4 now to

confirm that it's indeed the inverse

what we need to do is multiply matrix a

by the inverse of a and show that it's

equal to I 3 so in other words we need

to take matrix a which was 1 2 negative

1 negative 2 0 1 1 negative 1 0 and

multiply it by the inverse of that

matrix which is 1 1 2 and 1 1 1 2 3 4

and if this is the matrix of a if we did

it correctly we should get this answer 1

0 0 0 1 0 0 0 1 so let's find out if we

did it correctly so I'm gonna put my

answers in here first we need to take

the first row and multiply it by the

first column and add the products so

it's going to be 1 times 1 so we're

multiplying these two first and then

it's gonna be plus 2 times 1 that's

these 2

and then plus negative 1 times 2 so 1

times 1 is 1

2 times 1 is 2 negative 1 times 2 is

negative 2 these two cancel and so we

get 1 so we have the first entry now

let's multiply Row one by column two so

it's gonna be 1 times 1 plus 2 times 1

plus negative 1 times 3 so this is 1

plus 2 minus 3 1 plus 2 is 3 3 minus 3

is 0 now let's try room 1 by column 3 so

that's 1 times 2 plus 2 times 1 plus

negative 1 times 4 so this is 2 2 minus

4 2 plus 2 is 4 4 minus 4 is 0 now let's

move on to Row 2 and let's multiply it

by column 1

so it's gonna be negative 2 times 1 plus

0 times 1 plus 1 times 2 so this is

negative 2 plus 0 plus 2 negative 2 plus

2 is 0 now let's take Row 2 and multiply

it by column 2 so it's going to be

negative 2 times 1 plus 0 times 1 plus 1

times 3 so that's negative 2 plus 0 plus

3 negative 2 plus 3 is 1 now let's

multiply Row 2 by column 3 so negative 2

times 2 plus 0 times 1 plus 1 times 4 so

this is gonna be negative 4 plus 0 plus

4 which adds up to 0

now Row three times column one let's try

that so it's going to be 1 times 1 plus

negative 1 times 1 plus 0 times 2 so

it's 1 minus 1 plus 0 which is going to

be 0 next is going to be Row 3 and then

column 2 so this takes some time I mean

it's pretty laborious but that's how you

can confirm it so it's gonna be 1 times

1 plus negative 1 times 1 and then 0

times 3 so this is 1 minus 1 plus 0

which is 0

and finally the last one it's going to

be Row 3 times column 3 so that's going

to be 1 times 2 plus negative 1 times 1

plus 0 times 4 so it's 2 minus 1 plus 0

2 minus 1 is 1 and so we do get the

identity matrix so that tells us that

this is indeed the inverse of a so now

you know how to find the inverse of a

3x3 matrix

thanks for watching