The Inverse and Determinant of a Matrix · 矩阵的逆与行列式
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| inverse matrix/ɪnˈvɜːs ˈmeɪtrɪks/ | 逆矩阵 | nì jǔ zhèn |
| determinant/dɪˈtɜːmɪnənt/ | 行列式 | háng liè shì |
| identity matrix/aɪˈdentɪti ˈmeɪtrɪks/ | 单位矩阵 | dān wèi jǔ zhèn |
| not invertible/nɒt ɪnˈvɜːtɪbl/ | 不可逆的 | bù kě nì de |
Undoing a matrix
- Multiplying by a matrix transforms space — but can you undo it?
- For ordinary numbers, dividing by $5$ undoes multiplying by $5$.
- Matrices have their own "reverse" — the inverse matrix.
- First we need one number that decides whether a reverse even exists: the determinant.
撤销一个矩阵
- 乘以一个矩阵会变换空间——但你能撤销它吗?
- 对普通的数,除以 $5$ 撤销乘以 $5$。
- 矩阵有它们自己的"反向"——逆矩阵(inverse)。
- 首先我们需要一个数,来决定反向是否存在:行列式。
The determinant
- The determinant 行列式 of $\begin{bmatrix}a&b;\\c&d;\end{bmatrix}$ is the single number $ad - bc$.
- Geometrically, its size is the area of the parallelogram the columns span.
- A large determinant means the matrix stretches area a lot; a small one, very little.
- A determinant of zero means the columns line up — the parallelogram is flat.
行列式
- $\begin{bmatrix}a&b;\\c&d;\end{bmatrix}$ 的行列式(determinant)是一个数 $ad - bc$。
- 从几何上看,它的大小是列所张成的平行四边形的面积。
- 大的行列式意味着矩阵把面积拉伸很多;小的则很少。
- 行列式为零意味着列排成一条线——平行四边形被压平了。
For · 支持 $\begin{bmatrix}3 & 1\\ 1 & 2\end{bmatrix}$, the determinant is $3\cdot2 - 1\cdot1$. What is it? · 对于$\begin{bmatrix}3 & 1\\ 1 & 2\end{bmatrix}$，行列式是$3\cdot2 - 1\cdot1$。它是什么？
The determinant of $\begin{bmatrix}a&b;\\c&d;\end{bmatrix}$ is $ad - bc = 3\cdot2 - 1\cdot1 = 5$. · $\begin{bmatrix}a&b;\\c&d;\end{bmatrix}$的行列式是$ad - bc = 3\cdot2 - 1\cdot1 = 5$。
Geometrically, the size of a 2 × 2 determinant tells you… · 几何上，2 × 2行列式的大小告诉你……
The absolute value of the determinant is the area · 面积 scaled by the matrix — zero area means the columns are parallel. · 行列式的绝对值是矩阵缩放的面积——零面积意味着列是平行的。
The inverse
- The inverse matrix 逆矩阵 $A^{-1}$ is the matrix that undoes $A$.
- Multiplying them gives the identity matrix 单位矩阵 $I$ — the matrix version of $1$: $A A^{-1} = I$.
- For a $2\times2$ matrix, $A^{-1} = \dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a;\end{bmatrix}$.
- Notice the determinant $ad - bc$ sitting in the denominator.
逆矩阵
- 逆矩阵(inverse matrix)$A^{-1}$ 是撤销 $A$ 的那个矩阵。
- 把它们相乘得到单位矩阵(identity matrix)$I$——数字 $1$ 的矩阵版本:$A A^{-1} = I$。
- 对一个 $2\times2$ 矩阵,$A^{-1} = \dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a;\end{bmatrix}$。
- 注意行列式 $ad - bc$ 就坐在分母上。
Multiplying a matrix by its inverse · 反函数 gives the identity matrix. · 将矩阵与其逆相乘得到单位矩阵。
By definition $A A^{-1} = I$, the identity matrix — the matrix version of the number $1$. · 根据定义$A A^{-1} = I$，单位矩阵——数字$1$的矩阵版本。
When an inverse exists
- You cannot divide by $ad - bc$ if it is zero — so a zero determinant means no inverse.
- Such a matrix is not invertible 不可逆的: it squashes the plane onto a line and loses information.
- A non-zero determinant guarantees a unique inverse.
- So the determinant is the quick test: non-zero $\Rightarrow$ invertible.
逆何时存在
- 如果 $ad - bc$ 是零,你不能除以它——所以行列式为零意味着没有逆。
- 这样的矩阵是不可逆的(not invertible):它把平面压到一条线上,丢失了信息。
- 非零的行列式保证有唯一的逆。
- 所以行列式就是那个快速检验:非零 $\Rightarrow$ 可逆。
The determinant is an area factor · 行列式是面积因子
The determinant tells you how much a matrix scales area; here it scales area by four. · 行列式告诉你矩阵缩放面积的程度；在这里它将面积放大了四倍。
A 2 × 2 matrix has an inverse exactly when its determinant is… · 2 × 2矩阵有逆矩阵当且仅当其行列式是……
If $\det = 0$ the columns are parallel and no inverse exists — the matrix is not invertible. · 如果$\det = 0$，则列是平行的且不存在逆矩阵——该矩阵是不可逆的。
Select all · 所有 true statements. · 选择所有正确的陈述。
Only matrices with a non-zero determinant are invertible, so not every matrix has an inverse. · 只有行列式非零的矩阵才可逆，因此并非每个矩阵都有逆矩阵。
Why it matters
- An inverse solves a matrix equation $A\vec{x} = \vec{b}$ in one step: $\vec{x} = A^{-1}\vec{b}$.
- That is exactly how a system of linear equations is solved with matrices.
- The determinant also tells you when a system has no unique solution (det $= 0$).
- These ideas power computer graphics, cryptography, and data science.
它为什么重要
- 逆一步就解出矩阵方程 $A\vec{x} = \vec{b}$:$\vec{x} = A^{-1}\vec{b}$。
- 这正是用矩阵解线性方程组的方法。
- 行列式还告诉你一个方程组何时没有唯一解(det $= 0$)。
- 这些想法驱动着计算机图形学、密码学和数据科学。
A matrix has an inverse only when its determinant is non-zero. If $ad - bc = 0$, the formula divides by zero — there is no inverse, and a system $A\vec{x} = \vec{b}$ has either no solution or infinitely many. Always check the determinant first.
一个矩阵只有在它的行列式非零时才有逆。如果 $ad - bc = 0$,公式就除以零——没有逆,而方程组 $A\vec{x} = \vec{b}$ 要么无解,要么有无穷多解。总是先检查行列式。
Invert $A = \begin{bmatrix}3 & 1\\ 1 & 2\end{bmatrix}$.
- Determinant: $3\cdot2 - 1\cdot1 = 5$ (non-zero, so an inverse exists).
- Swap the diagonal, negate the off-diagonal, divide by $5$:
$$A^{-1} = \frac{1}{5}\begin{bmatrix}2 & -1\\ -1 & 3\end{bmatrix}.$$
- Check: $A A^{-1} = I$. ✓
求 $A = \begin{bmatrix}3 & 1\\ 1 & 2\end{bmatrix}$ 的逆。
- 行列式:$3\cdot2 - 1\cdot1 = 5$(非零,所以逆存在)。
- 交换对角线,给副对角线取负,再除以 $5$:
$$A^{-1} = \frac{1}{5}\begin{bmatrix}2 & -1\\ -1 & 3\end{bmatrix}.$$
- 检验:$A A^{-1} = I$。✓
The determinant $ad - bc$ is one number measuring the area a matrix scales. The inverse $A^{-1}$ undoes the matrix, with $A A^{-1} = I$ (the identity matrix). An inverse exists only when the determinant is non-zero; a zero determinant means the matrix is not invertible.
行列式 $ad - bc$ 是一个数,度量矩阵缩放的面积。逆矩阵 $A^{-1}$ 撤销这个矩阵,满足 $A A^{-1} = I$(单位矩阵)。逆只有在行列式非零时才存在;行列式为零意味着矩阵不可逆。