Finding inverses and determinants
Rule of Sarrus of Determinants A alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus)
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- I don't want to beat a dead horse by showing you all of
- the different ways to find a determinant, but it might be
- useful to beat this dead horse because you'll see it done in
- different ways, in different context.
- And I thought I would at least show you that what we have
- been covering, so far, is very consistent with a way of
- determining, or finding determinants, that you might
- have been exposed to in your Algebra Two class.
- It's called the rule of Sarrus.
- Let me just prove it for you.
- Let's say we want to find a determinant.
- Let's say you want to find this determinant.
- So our matrix is a, b, c, d, e, f, g, h, i.
- We know how to do this.
- This is equal to, let's just go down that first row, a
- times the determinant of e, f, h, i minus b times the
- determinant of d, g, f, i plus c times the
- determinant of d, e, g, h.
- And what are these equal to?
- This is going to be equal to a, so let me write this, a
- times ei minus fh.
- And this is going to be minus b times dI minus fg.
- This is going to be plus c times dh minus eg and if we
- multiplied this out we get this is equal to aei minus afh
- minuses bdi plus right, minus times a minus, plus bfg plus
- cdh minus ceg.
- Now let me group the positive and the negative terms.
- So this term is positive, this term is positive and that term
- is positive.
- So we have this being equal to aei plus bfg plus cdh.
- Those are our positive terms.
- And then our negative terms are here.
- We have that term, that term, and that term.
- So we have minus afh minus bdi minus ceg.
- So this is a formula for the determinant of this matrix
- right here.
- Let's see what it actually looks like.
- Let me rewrite it.
- Let me rewrite our matrix.
- We do it in green.
- So we have a, b, c, d, e, f, g, h, i.
- We wanted to find its determinant.
- So let me show you something interesting here.
- aei is what?
- aei is a product of this guy, this guy, and that guy.
- So, you're essentially going along that
- diagonal right there.
- Now what is bfg?
- You're going this guy, this guy, and then you're going all
- the way down to this guy.
- So it's like if you imagine that when you come out of this
- side, you come out of this side.
- There's some video games where you go out one end and you end
- up showing up on the other end like that.
- It would also be a diagonal.
- Or even a better way to visualize it, let me redraw
- these two columns.
- Let me augment this determinant.
- It's not official terminology, but I think you'll get what
- I'm trying to do.
- So if I write these first two columns again. a, d,
- g, and b, e, h.
- This guy right here bfg, it's this one right here, this
- diagonal right there.
- And then you might guess what's about to happen.
- Where is cdh?
- It's this diagonal.
- It's that diagonal right there.
- So you take this product, add it to this product, add it to
- this product.
- And then you subtract these guys.
- Now what are these guys?
- Where is the afh?
- That one right there.
- So you subtract out afh, and then you subtract out bdi.
- bdi is that one right there.
- And then you have ceg, which is this one right there.
- So the Rule of Sarrus, sounds like something in
- The Lord of the Rings.
- The Rule of Sarrus is essentially a quick way of
- memorizing this little technique.
- You write the two columns again, you say, ok, this
- product plus this product plus this product, minus this
- product minus this product minus that product.
- Let's actually do it with the 3 by 3 matrix to make it clear
- that the Rule of Sarrus can be useful.
- So let's say we have the matrix, we want the
- determinant of the matrix, 1, 2, 4, 2, minus 1, 3, and then
- we have 4, 0, minus 1.
- We want to find that determinant.
- So by the Rule of Sarrus, we can rewrite
- these first two columns.
- So 1, 2, 2, minus 1, 4, 0.
- We rewrote those first two columns.
- And to figure out this determinant we take this guy.
- What is this going to be?
- 1 times minus 1 times minus 1.
- That is just a 1.
- Right, the minuses cancel out.
- Plus this guy, plus this product right here.
- I should draw a little bit neater.
- So what is this?
- 2 times 3 times 4.
- 2 two times 3 is 6.
- 6 times 4 is 24, plus 24.
- And then we take this guy right here.
- 4 times 2 times 0, anything times 0 is a 0.
- So that's going to be plus 0.
- And then we subtract out these guys.
- So you have 4 times 4, times minus 1.
- That's minus 16.
- It's minus 16, but we're going to be on the
- minus side of things.
- So it's 4 times minus 1 times 4, is minus 16.
- But since were going to do a minus on it, it's
- going to be plus 16.
- So it's 16.
- Then you have a 0 times 3 times 1.
- That of course is going to be 0.
- Would be a minus 0, but we can ignore it.
- So we can say plus 0 or minus 0 same thing.
- Then you have a minus 1 times 2 times 2.
- So that's 4 times minus 1 which is minus 4.
- When you go in this direction, from the top right to the
- bottom left, you are subtracting.
- So this would be a minus 4 but since we're subtracting, this
- becomes a plus 4.
- So the value of our determinant is equal to, by
- the Rule of Sarrus, we're going to have
- 16 plus 4 is a 20.
- 20 plus 25 which is equal to 45.
- So that actually is, I'd have to say, a faster way of
- computing this 3 by 3 derivative.
- And I just want to show you this is completely equivalent
- to the definition that I introduced you to a couple
- videos ago.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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