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

也即下角行列式，还有对角行列式。

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

一旦和j帽完美地对齐 行列式就零。

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

这就对二维行列式理解。

Namely, the determinant of our transformation matrix.

即变换矩阵行列式。

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

下角行列式就，主对角线以元素都为零行列式。

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

角行列式就，主对角线以下元素都为零行列式。

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 行列式不为零。

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

但（！），这行列式一个关键概念，所有这些面积都会被放大相同倍数。

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

零 从对角线元素中减去lambder然后寻找行列式为零时候。

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

n 阶行列式定义，这就关于一个全新公式了，里面有加总符号，还有逆序数符号。

That is if it has zero determinant.

那就说如果它有零行列式。

Then you compute the determinant of this matrix.

然后计算这个矩阵行列式。

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

所以它面积将变换行列式乘以那个值。

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后面要学 n 阶行列式定义。

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

行列式很重要 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 每当空间方向颠倒 行列式就负。

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

阶行列式计算没有什么技巧，就对角线法则，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， 给这个常数起了一个名字，它被称为变换行列式。

Now, in principle Alice could  compute this determinant.

现在，原则爱丽丝可以计算这个行列式。

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

行列式绝对值 仍然告诉你面积被缩放因子。