In the last video, we stumbled

upon a way to figure out the inverse for an invertible

matrix. So, let’s actually use

that method in this video right here. I’m going to use the same matrix

that we started off with in the last video. It seems like a fairly

good matrix. We know that it’s reduced row

echelon form is the identity matrix, so we know

it’s invertable. So, let’s find its inverse. The technique is pretty

straightforward. You literally just apply the

same transformations you would apply to this guy to get you

to the identity matrix, and you would apply those same

transformations to the identity matrix. That’s because the collection

of those transformations, if you represent them as matrixes,

are really just the inverse of this guy. Let’s just do it. So I’ll create an augmented

matrix here. Maybe I’ll do it right here. Let me make it a little

bit neater. First, I’ll write a. It’s 1, minus 1, 1. And then minus 1, 2, 1. Minus 1, 3, 4. And then I’ll augment it with

the identity matrix, with 1, 0, 0, 0, 1, 0, 0, 0, 1. Now, if I want to get a into

reduced row echelon form, maybe I’ll replace

the second row. I’ll keep the first row

the same for now. Let me just draw it like this. The entire first row:

1, minus 1, minus 1. It’s going to be augmented

with 1, 0, 0. Keep the whole first

row the same. Let’s replace the second

row with the second row plus the first row. Minus 1 plus 1 is 0. 2 plus minus 1 is 1. 3 plus minus 1 is 2. 0 plus 1 is 0. 1 plus– oh, sorry. That was a tricky one. 0 plus 1 is 1. 1 plus 0 is 1. 0 plus 0 is 0. All I did is I added

these two rows up. Now, this third row. Let me replace– I want

to get a zero here. Let me replace the third

row with the third row minus the first row. 1 minus 1 is 0. 1 minus minus 1 is 2. 4 minus minus 1 is 5. 0 minus 1 is minus 1. 0 minus 0 is 0. And then 1 minus 0 is 1. Just like that. Now, what do we want to do? Well, we’ve gotten this far. We want to zero out that

guy and that guy. Let’s keep our second

row the same. Let me write it down here. It’s 0, 1, 2, and then you

augmented it with 1, 1, 0. Just like that. And let’s replace my first row

with the first row plus the second row. 1 plus 0 is 1. Minus 1 plus 1 is 0. That’s why I did that,

to get a zero there. Minus 1 plus 2 is 1. 1 plus 1 is 2. 0 plus 1 is 1. 0 plus 0 is 0. And now, I also want to zero

out this guy right here. Let’s replace the third row

with the third row minus 2 times the second row. 0 minus 2 times 0 is 0. 2 minus 2 times 1 is 0. 5 minus 2 times 2 is 5

minus 4, that’s 1. Minus 1 minus 2 times 1–

that’s minus 1 minus 2– is minus 3. 0 minus 2 times 1,

that’s minus 2. And then, 1 minus 2 times

0 is just 1 again. All right, home stretch. Now, I just want to zero out

these guys right here. All right, so just let me keep

my third row the same. Let me switch colors, keep

things colorful. It’s 0, 0, 1. We’re going to augment it with

minus 3, minus 2, and 1. Now, let’s replace our first row

with the first row minus the third row. 1 minus 0 is 1. 0 minus 0 is 0. 1 minus 1 is 0. 2 minus minus 3, that’s 5. 1 minus minus 2 is 3. 0 minus 1 is minus 1. Now, let’s replace the second

row with the second row minus 2 times the third row. 0 minus 2 times 0 is 0. 1 minus 2 times 0 is 0. 2 minus 2 times 1 is– I’m

sorry, I just– oh, whoops. Let me– we have to be very

careful not to make any careless mistakes. 0 minus 2 times 0 is 0. 1 minus 2 times 0 is 1. It’s not 0. 2 minus 2 times 1 is 0. 1 minus 2 times minus

3– that is 1 plus 2 times 3– that is 7. 1 minus 2 times minus 2, that’s

1 plus 4, which is 5. And then, 0 minus 2 times

1, so that’s minus 2. And just like that, we’ve

gotten the A part of our augmented matrix into reduced

row echelon form. This is the reduced row

echelon form of A. And when you apply those exact

same transformations– because if you think about it, that

series of matrix products that got you from this to the

identity matrix– that, by definition, is the

identity matrix. So you apply those same

transformations to the identity matrix, you’re going

to get the inverse of A. This right here is A inverse. And we have solved for the

inverse, and it actually wasn’t too painful.

first view and first rate

Thank you Sal , great video as alwaysssssss

With the inverse matrix (A) you can transform a matrix (B) that is relative to the matrix(A) to the World coordinate system or the parent matrix of (A)

You have an error at the last matrix. one – 2 * – three = one – six = -five

Damn youtube filter..

1 – (2 times -3) = 1- (-6) =7

oops sorry I was thinking like the power of.. ) – with – = + 🙂

lol owned yourself :p

i also mentioned that in the previous video; you should make an example with row permutations (in case of a pivot misplacement)

@minxshania Generally speaking, though I'm not sure of it, row operations executed on any matrix dedicated to manipulating it into the form of the identity matrix are, as I understand, basically simply decided intuitively in intention to turn all the cells in each row to 0 except for a single cell, doing so row by row, each one at a time. Additionally, though I know no other suitable essentially different explanation, I pretty logically suppose there exists such one.

Jordan

@TGBProductionsz Most likely in Pre-Calculus. I'm a Sophomore in high school and taking Pre-Calculus, which about the earliest you can take Pre-Calculus. I've heard of Freshmen taking Pre-Calculus, but you would most likely take it during your Senior year of high school, if you don't take advanced math.

you are a life saver 😀

Thank you very much

4 years from now.

confuse 🙁

Im in 7th grade and in algebra

thanks!

Greaatt!! finally, I understand it !! Awesome!

I'm in second year University, reviewing Linear Algebra and Vector Spaces. Nice to meet you. 🙂

I really want to know what this guy could have asked, that was so offensive that it got too many negative downvotes, but his answer was 7.

Gold!

does A have to end up as an identity matrix when put into reduced row echelon form for it to have an inverse?

What is the name of the method you used to find inverse of the matrix????

thanks. very good explanation

Best video still

Can't you multiply a row by a number like ' R1*2?!