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Then upper triangular, everything below the diagonal is a 0, and everything above the diagonal probably isn't a 0, but you never know.

So the transformation of x2 is equal to our transformation matrix 1, 2, minus 1, 0 times 2, minus 2. And so this will be equal to 1 times 2 is 2. Plus minus 1 times minus 2.

1, 2, minus 1, 0 times minus 2 minus, minus 2. And so what would this be equal to?

This just comes from the definition of a 2 by 2 determinant.

2 minus 2 times 1 is 0.

Right by twos!

I'm ignoring the zero because it's in the same column as the 1, 2, 0 0, 2, 3, and then 2, 0, 0.

1 minus 2 minus so it's 1 minus 2 plus minus 1 times minus 2. That's going to be plus 2.

We defined the determinant of a 2 by 2 matrix not in terms of a determinant.

0 minus 2 times 1, that's minus 2.

2 minus 2, times 1, is 0.

5 minus 2, times 2, is 1.

Two by two. Right and left.

Let's take our transformation of x2.

So it's 1, 0, 2 0, 1, 2 2, 3, 0, just like that.

But the motivation comes from the exact same place as the 2 by 2 version.

So far we've been able to define the determinant for a 2 by 2 matrix.

And then, 0 minus 2 times 1, so that's minus 2.

0 minus 2 is minus 2.

And then have 2 times 2 is 4. Plus 0 times minus 2. So it's 4, 4.

In the last video we defined the notion of a determinant of a 2 by 2 matrix.

I've got a 2 by 2 matrix here. Let's say A is equal to a, b, c, d. So I'm going to keep it really general.

So times the determinant of the matrix a2 2, a2 3, a3 2, and then a3 3. Just like that.

Well, you keep doing it, and you're going to get all the way down to a 2 by 2 matrix.

And then 2 times minus 2 is minus 4. Plus 0.

Then minus 2 times get rid of that row, that column 0, 3, 2, 0. And then plus 0 times 0, 2, 2, 0. That's this one right there.

Minus 2 times the determinant. You cross these guys. Cross that row and column out.

0 minus 2 is minus 2.

This is 36 times 2.

The upper triangular everything below the diagonal.

So we can say plus 0 or minus 0 same thing.

So this matrix right here.

And then we have 2 times minus 2, which is minus 4. 2 times minus 4. And then plus 0 times, so it's minus 4.

So we'll get a 0, we'll get a 1 minus 2, times 0, and we'll get a 2 minus 2, times 1. So that's a 0.

Now, we've defined or we've calculated or we've defined our determinant of this matrix in terms of just a bunch of 2 by 2 matrices.

at a time.

Don't give them four, give them two plus two.

Minus 2 times minus 2 is 4.

Two!

Two.

Two.

The transformation of x1 is equal to 1, 2, minus 1, 0 times minus 2, 2. So what is this equal to?

Permission to speak on the motion shall be accorded only to two representatives in favour of and to two opposing the division.

In this case, our base case is the case of a 2 by 2 matrix.