计算 matrix operations
Perform 矩阵 addition, multiplication, determinant, inverse, 和 transpose. 必要为 线性 algebra, graphics, 和 ML.
- 1
Matrix 操作: addition (元素-wise, 相同 dimensions), multiplication (dot product 的 rows 和 columns, [m×n] × B[n×p] = C[m×p]), transpose (swap rows 和 columns), determinant (标量为 square matrices, 零 means singular/non-invertible), inverse (^(-1) such 该 × ^(-1) = I, exists 仅如果 det ≠ 0).
- 2 矩阵乘法 →
Multiply two matrices. Enter dimensions 和 values. tool computes dot product 的每个行 的具有每个列 的 B. Matrix multiplication 是不 commutative (×B ≠ B× 一般而言) 和是 O(n^3) 为 n×n matrices.
- 3 矩阵行列式计算器 →
计算 the determinant of a square matrix. The determinant indicates whether a matrix is invertible (det ≠ 0) and the scaling factor of the linear transformation. For 2×2: ad-bc. For larger matrices, use LU decomposition or cofactor expansion.
- 4 矩阵求逆计算器 →
计算 the inverse of a square matrix. The inverse exists only if the determinant is non-zero. The inverse is used to solve linear systems (Ax = b → x = A^(-1)b). For large matrices, use Gaussian elimination instead of computing the inverse explicitly.
常见问题
- 当是 矩阵 invertible?
- square 矩阵是 invertible 如果和 仅如果其 determinant 是 non-零. Geometrically, 矩阵 represents transformation 该不 collapse space 到下 维度. 如果 det = 0, transformation 是 singular (collapses dimensions) 和 cannot 是 reversed.
- 为什么不 总是 use inverse 到 solve Ax = b?
- Computing inverse 是 O(n^3) 和 numerically 不稳定 (rounding 错误 accumulate). Solving via Gaussian elimination 或 LU decomposition 是 also O(n^3) 但更多稳定和 faster (下常量因子). Use inverse 仅当 你 need 到 solve Ax = b 为许多不同 b 具有相同。