That is, our upper and lower triangular determinants and diagonal determinants.

也即我们的上下三角行列，还有角行列。

Once I had lines up perfectly with j hat, the determinant is zero.

一旦我和j帽完美地齐 行列零。

So that's the understanding of determinants in two dimensions.

这二维行列的理解。

Namely, the determinant of our transformation matrix.

即我们的变换矩阵的行列。

A lower triangular determinant is a determinant in which the elements above the main diagonal are all zero.

下三角行列，角线以上的元素都为零的行列。

An upper triangular determinant is a determinant in which the elements below the main diagonal are all zero.

上三角行列，角线以下的元素都为零的行列。

It means the basis vectors continue to span the full two dimensions of space, and the determinant is not zero.

它意味着基向量继续张成整个二维空间 行列不为零。

But(! ) , and this is a key idea of determinants, all these areas get scaled by the same amount.

但（！），这行列的一个关键概念，所有这些面积都会被放大相同的倍数。

Zero. Subtract off lambder from the diagonal elements and look for when the determinant is zero.

零 从角线元素中减去lambder然后寻找行列为零的时候。

The definition of an n-order determinant is about a completely new formula. There are summation symbols and inverse order number symbols.

n 阶行列的定义，这关于一个全新的公了，里面有加总符号，还有逆序数符号。

That is if it has zero determinant.

那说如果它有零行列。

Then you compute the determinant of this matrix.

然后计算这个矩阵的行列。

So its area will be the determinant of the transformation multiplied by that value.

所以它的面积将变换的行列乘以那个值。

The definition of an n-order determinant is introduced by full permutation and inverse order number.

由全排列和逆序数引出我们后面要学的 n 阶行列的定义。

It's not just a coincidence that the determinant is once again important.

行列很重要 这不巧合。

Whenever this happens, whenever the orientation of space is inverted, the determinant will be negative.

每当这种情况发生 每当空间的方向颠倒的 行列负的。

The calculation of the third-order determinant has no skills. It is just the diagonal rule. After calculation, it's done.

三阶行列的计算没有什么技巧，角线法则，算完事了。

More specifically we have a name for that constant, it's called the determinant of the transformation.

更具体地说， 我们给这个常数起了一个名字，它被称为变换的行列。

Now, in principle Alice could  compute this determinant.

现在，原则上爱丽丝可以计算这个行列。

The absolute value of the determinant, though still tells you the factor by which areas have been scaled.

行列的绝值 仍然告诉你面积被缩放的因子。