Berechnen matrix operations
Perform matrix addition, multiplication, determinant, inverse, und transpose. Essential für linear algebra, graphics, und ML.
- 1
Matrix operations: addition (element-wise, same dimensions), multiplication (dot product of rows and columns, A[m×n] × B[n×p] = C[m×p]), transpose (swap rows and columns), determinant (scalar for square matrices, zero means singular/non-invertible), inverse (A^(-1) such that A × A^(-1) = I, exists only if det ≠ 0).
- 2 Matrix-Multiplikation →
Multiply two matrices. Enter the dimensions and values. The tool computes the dot product of each row of A with each column of B. Matrix multiplication is not commutative (A×B ≠ B×A in general) and is O(n^3) for n×n matrices.
- 3 Matrix-Determinante →
Berechnen 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 Matrix-Inverse →
Berechnen 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.
Häufige Fragen
- When is a matrix invertible?
- A square matrix is invertible if and only if its determinant is non-zero. Geometrically, the matrix represents a transformation that does not collapse space to a lower dimension. If det = 0, the transformation is singular (collapses dimensions) and cannot be reversed.
- Why not always use the inverse to solve Ax = b?
- Computing the inverse is O(n^3) and numerically unstable (rounding errors accumulate). Solving via Gaussian elimination or LU decomposition is also O(n^3) but more stable and faster (lower constant factor). Use the inverse only when you need to solve Ax = b for many different b with the same A.