🧮 伴随矩阵计算器Adjoint (Adjugate) Matrix Calculator
输入 2×2 到 6×6 的方阵，精确计算伴随矩阵 adj(A)（余子式矩阵的转置）、余子式矩阵和行列式，并在可逆时给出分数形式的逆矩阵 A⁻¹ = adj(A) / det(A)。整数和最多 6 位小数都按精确有理数计算。Enter a square matrix from 2×2 to 6×6 to get its adjugate adj(A) — the transpose of the cofactor matrix — exactly, together with the cofactor matrix, the determinant and, when it exists, the inverse A⁻¹ = adj(A) / det(A) as fractions. Whole numbers and decimals with up to 6 places are handled as exact rationals.
使用说明与常见问题How-to & FAQ
「伴随矩阵计算器」是什么？What is the Adjoint (Adjugate) Matrix Calculator tool?
输入 2×2 到 6×6 的方阵，精确计算伴随矩阵 adj(A)（余子式矩阵的转置）、余子式矩阵和行列式，并在可逆时给出分数形式的逆矩阵 A⁻¹ = adj(A) / det(A)。整数和最多 6 位小数都按精确有理数计算。Enter a square matrix from 2×2 to 6×6 to get its adjugate adj(A) — the transpose of the cofactor matrix — exactly, together with the cofactor matrix, the determinant and, when it exists, the inverse A⁻¹ = adj(A) / det(A) as fractions. Whole numbers and decimals with up to 6 places are handled as exact rationals.
如何使用「伴随矩阵计算器」？How do I use the Adjoint (Adjugate) Matrix Calculator tool?
在「伴随矩阵计算器」中，填写上方字段等参数，结果会随输入实时计算并显示在下方。In the Adjoint (Adjugate) Matrix Calculator: Fill in the fields above; results are recalculated instantly as you type.
什么是伴随矩阵？What is the adjugate (adjoint) matrix?
把方阵 A 每个元素 aᵢⱼ 换成它的代数余子式 Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ（Mᵢⱼ 是删去第 i 行第 j 列后的行列式），得到余子式矩阵 C；再转置，就是伴随矩阵 adj(A) = Cᵀ。它满足 A · adj(A) = det(A) · I。Replace each entry aᵢⱼ of a square matrix A with its cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ, where Mᵢⱼ is the determinant left after deleting row i and column j. That gives the cofactor matrix C; its transpose is the adjugate adj(A) = Cᵀ, which satisfies A · adj(A) = det(A) · I.
和逆矩阵有什么关系？How does it relate to the inverse?
当 det(A) ≠ 0 时，A⁻¹ = adj(A) / det(A)，本工具以最简分数给出每个元素；det(A) = 0 时矩阵奇异、没有逆矩阵，但伴随矩阵仍然存在。注意「伴随矩阵」在线性代数里有时也指共轭转置（埃尔米特伴随），这里算的是古典伴随（adjugate）。When det(A) ≠ 0, A⁻¹ = adj(A) / det(A), shown here with each entry as a fraction in lowest terms. When det(A) = 0 the matrix is singular and has no inverse, but the adjugate still exists. Note that "adjoint" can also mean the conjugate transpose (Hermitian adjoint); this tool computes the classical adjoint, the adjugate.
结果是精确的吗？Are the results exact?
是。小数先按 10 的幂放大成整数，再用 BigInt 和 Bareiss 无除法余数消元求行列式，最后再缩回，因此不会有浮点舍入误差。每个数最多 6 位小数、绝对值不超过 1,000,000，矩阵最大 6×6。Yes. Decimals are scaled up to whole numbers by a power of ten, determinants are found with BigInt using fraction-free Bareiss elimination, and the result is scaled back — so there is no floating-point rounding. Each number may have up to 6 decimal places and be at most 1,000,000 in absolute value; the largest matrix is 6×6.
「伴随矩阵计算器」免费吗？需要注册吗？Is the Adjoint (Adjugate) Matrix Calculator free? Do I need an account?
「伴随矩阵计算器」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 Adjoint (Adjugate) Matrix Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.