⁻¹ 矩阵求逆Invert Matrix
求方阵的逆：把输入的整数、小数或分数当作精确有理数，用高斯–约当消元精确求逆，再按需显示为小数或分数。本地处理。Invert a square matrix: integers, decimals and fractions are read as exact rationals and inverted exactly with Gauss–Jordan elimination, then shown as decimals or fractions. Processed locally.
使用说明与常见问题How-to & FAQ
「矩阵求逆」是什么？What is the Invert Matrix tool?
求方阵的逆：把输入的整数、小数或分数当作精确有理数，用高斯–约当消元精确求逆，再按需显示为小数或分数。本地处理。Invert a square matrix: integers, decimals and fractions are read as exact rationals and inverted exactly with Gauss–Jordan elimination, then shown as decimals or fractions. Processed locally.
如何使用「矩阵求逆」？How do I use the Invert Matrix tool?
在「矩阵求逆」中，把内容粘贴或输入到上方输入框，按需设置「显示」，结果会实时显示在下方，点击「复制结果」即可复制。In the Invert Matrix: Paste or type into the input box, adjust "Show as"; the result appears below instantly — click "Copy result" to copy it.
结果精确吗？Are the results exact?
是。整数、小数（如 1.0000000001）、科学计数（1e-3）和分数（1/2）都按精确有理数读入，消元全程用大整数（BigInt）分数计算，不经过浮点，所以不会出现舍入误差。显示为小数时，有限小数原样显示，其他值四舍五入到 12 位有效数字并在下方注明；选「分数」可看到精确值，如 1/3。Yes. Integers, decimals (such as 1.0000000001), scientific notation (1e-3) and fractions (1/2) are read as exact rationals, and the elimination runs entirely on big-integer (BigInt) fractions with no floating point, so there is no rounding error. In decimal display, terminating decimals are shown in full and other values are rounded to 12 significant digits with a note below; choose Fractions to see exact values such as 1/3.
什么是病态矩阵？What is an ill-conditioned matrix?
条件数很大的矩阵（例如 1 1 / 1 1.0000000001，条件数约 4×10¹⁰）：输入的微小改动会让逆矩阵大幅变化。条件数超过 10⁶ 时页面会给出提示；因为这里是精确计算，结果本身仍然准确，例如上面的例子得到 10000000001 −10000000000 / −10000000000 10000000000。A matrix with a very large condition number (for example 1 1 / 1 1.0000000001, condition number about 4×10¹⁰): tiny changes to the input change the inverse a lot. Above 10⁶ the page shows a warning; since the computation is exact the result itself is still correct, e.g. the example above gives 10000000001 −10000000000 / −10000000000 10000000000.
怎么输入？How do I enter the matrix?
每行一行，数之间用空格或逗号分隔，也可以用分号分行；最多 20×20。每行个数不同、不是方阵、或含有不是数字的内容时，会分别指出是哪一行哪个数。行列式为 0 的矩阵不可逆。One row per line, numbers separated by spaces or commas; a semicolon also starts a new row. Up to 20×20. Rows of different lengths, a non-square matrix or a non-numeric entry each get a message naming the row and entry. A matrix with determinant 0 has no inverse.
输入和输出是什么样的？What does the input and output look like?
例如在「矩阵求逆」中输入 1 0 ⏎ 0 1，输出为：1 0 ⏎ 0 1（⏎ 表示换行）。For example, the Invert Matrix turns the input 1 0 ⏎ 0 1 into: 1 0 ⏎ 0 1 (⏎ marks a line break).
有哪些选项可以调整？What options can I adjust?
「矩阵求逆」的设置：「显示」可选 小数、分数（精确）（默认 小数）。每次生成结果都按当前设置处理。Settings in the Invert Matrix — "Show as": Decimals, Fractions (exact) (default Decimals). The result always uses the current settings.
「矩阵求逆」免费吗？需要注册吗？Is the Invert Matrix free? Do I need an account?
「矩阵求逆」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Invert Matrix is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.