Matrix Operations
Matrix Operation Rules
Addition/Subtraction: Matrices must have the same dimensions. Multiplication: Number of columns in A must equal number of rows in B.
Determinant Calculator
2×2: det(A) = ad - bc
3×3: det(A) = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)
Eigenvalues & Eigenvectors
Characteristic Polynomial: det(A - λI) = 0
For 2×2: λ² - tr(A)λ + det(A) = 0
Eigenvalue Properties
Sum of eigenvalues = trace(A). Product of eigenvalues = det(A). For symmetric matrices, all eigenvalues are real.
System of Linear Equations
Solution Methods
Gauss Elimination: Transform to row echelon form. Cramer's Rule: Use determinants (works when det(A) ≠ 0).
Matrix Decompositions
LU: A = LU (L: lower triangular, U: upper triangular)
QR: A = QR (Q: orthogonal, R: upper triangular)
Spectral: A = PDP⁻¹ (P: eigenvectors, D: eigenvalues)