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Here's the two by two identity matrix, here's the three by three identity matrix, here's the four by four identity matrix.

But it's the unit circle.

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So if I have a unit circle like that.

And once again, this is a unit circle, so it has a radius of 1.

Because once again, this is the unit circle.

But when B is the identity matrix, this does hold true.

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And only one contrary data point can blow your theory out of the water.

So it turns out that in the space of matrices as an identity matrix as well.

And the special matrices, like the identity matrix I want to tell you about, in the next and final video now linear algebra review.

In mathematics, a unit circle is a circle with a radius of one.

So the identity matrix, has the property that it has ones along the diagonals, right, and so on and is zero everywhere else, and so, by the way the one by one identity matrix is just a number one.

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Let me try to really fast draw a new unit circle.

If we have a unit circle, we define

Well it's 1, because this is a unit circle and this is the radius of the unit circle.

Well the hypotenuse is this side, which has length 1 because it was the unit circle and that's the radius of it.

This 3 carbon chain, where everything is a single bond, is propane.

But the unit circle kind of extends that SOH CAH TOA definition.

It tells us if we have a unit circle And this is a picture of a unit circle here. A unit circle is just a circle centered at the origin, centered at the point 0, 0 , and it has a radius of 1.

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So I subscript n by n is the n by n identity matrix.

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So I got the unit circle right over here now clearly when we're at forty five degrees sine and cosine are equal to each other or forty five degrees is the same thing as pi over four there they're equal to each other you might be tempted to do this over here but here the cosine is negative sine is positive. so that won't work they're both negative over here but they're equal. so that's another value and then this won't work

I times A equals A times I equals A. Just make sure we have the dimensions right, so if A is a n by n matrix, then this identity matrix that's an m by n identity matrix.

In the last presentation we kind of re defined the sine, the cosine, and the tangent functions in a broader way where we said if we have a unit circle and our theta is, or our angle, is let me use the right tool let's say, and our angle is the angle between, say, the x axis and a radius in the unit circle, and this is our radius. the coordinate of the point where this radius intersects the unit circle is x comma y.

This is one by one matrix just and it's not a very interesting identity matrix and informally when I or others are being sloppy, very often, we will write the identity matrix using fine notation.

Think of it as basically the inverse of A and that's the inverse of A and second set, you know, 10 equals p of A and of temp times A. This is indeed the identity matrix with essentially ones on the diagonals and zeros on the off diagonals, up to a numerical round off.

And if A is m by n then this identity matrix, right, for matrix multiplication make sense that has a m by n matrix because this m has a match up that m And in either case the outcome of this process is you get back to Matrix A, which is m by n.