So, create a matrix whose first column is the same as that of our matrix, and whose second column is the output vector, and take its determinant.

所以， 创建个矩阵， 其第列与我们的矩阵相同，第二列是输出向量，然后取其行列式。

Similarly, for three equations and three unknowns, there will be no inverse if the corresponding transformation squishes 3D space onto the plane, or even if itsquishes it onto a line or a point, those all correspond to a determinant of zero.

对于三个方程和三个未知数 如果相应的变换将三维空间挤压到平面上 或者即使它挤压到条线或点上 也不会有逆 这些都对应于行列式为零。

In the process of eliminating the unknowns, we can obtain the conversion form of the determinant. Later, we call this Cramer's rule.

在将未知数消掉的过程，我们就可以得出行列式的转换形式了，后期我们称之为克拉默法则。

That is if it has zero determinant.

那就是说如果它有零行列式。

So if you didn't see chapter five of this series, where I talk about the determinant, now would be a really good time to go take a look.

如果你没有过这个系列的第五章 我讲了行列式 现在是个很好的时间。

A diagonal determinant is a special form of upper and lower triangular determinants. Because you can tacitly assume that it satisfies both upper and lower triangular determinants.

对角行列式就是上下三角行列式的种特色形式，因为你可以把它默认为上下三角行列式它都满足。

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

所以它的面积将是变换的行列式乘以那个值。

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

每当这种情况发生 每当空间的方向是颠倒的 行列式就是负的。

The transformed version of this guy is spanned by the output vector and the second column of the matrix, and its area will have been multiplied by the determinant of the matrix.

这个变换后的版本由输出向量和矩阵的第二列张成，其面积将乘以矩阵的行列式。

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

三阶行列式的计算没有什么技巧，就是对角线法则，算完事了。

The second-order determinant, or the use of the determinant, is to calculate some complex multivariate linear equations. Because sometimes it is quite difficult to calculate when the number of elements increases.

二阶行列式，或者说行列式的用处，就是把些复杂的多元线性方程组计算出来，因为有时候元变多了还是蛮难计算的。

Actually, to be more accurate, you should think of the signed area of this parallelogram, in the sense described by the determinant video.

实际上， 为了更准确理解， 你应该考虑这个平行四边形的有符号面积，就像行列式视频描述的那样。

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

行列式的绝对值 仍然告诉你面积被缩放的因子。

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 阶行列式的定义。

So for this video, I'm assuming that everybody has watched Chapter five on the Determinant, and chapter seven, where I introduced the idea of duality.

在这个视频 我假设每个人都过第五章的行列式 第七章 我介绍了对偶的概念。

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

但是（！），这是行列式的个关键概念，所有这些面积都会被放大相同的倍数。

Given a three by three matrix, e.g., it seems a lot harder for a solution to when it squishes space onto a line, compared to when it squishes things onto a plane, even though both of those are zero determinant.

给定个3×3矩阵 例如 当它把空间挤压到条直线上时 它的解乎要比它把东西挤压到个平面上的解困难得多 即使它们的行列式都是零。

First is the first knowledge point, second-order and third-order determinants. These two determinants are relatively simple and basic. We will not go into too much detail about the calculation methods.

首先是第个知识点，二阶与三阶行列式，这两个行列式比较简单，比较基础，我们不过多赘述计算方法。

What's more, if V is on the left of W, it means that orientation was flipped during that transformation, which is what it means for the determinant to be negative.

更重要的是 如果V在W的左边 这意味着在变换过程方向被翻转了 这意味着行列式为负。

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

更具体说， 我们给这个常数起了个名字，它被称为变换的行列式。