Linear Algebra Helper
Linear Algebra Helper
Linear Algebra Results
Matrix A Dimensions
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Matrix B Dimensions
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Operation
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Compatible: –
Detailed Solution
Linear Algebra Concepts & Formulas
Core Matrix Operations:
Key Formulas:
1. Matrix Addition: C[i,j] = A[i,j] + B[i,j]
2. Matrix Multiplication: C[i,j] = Σ A[i,k] × B[k,j]
3. Determinant (2×2): det = a₁₁×a₂₂ - a₁₂×a₂₁
4. Determinant (3×3): Sarrus rule or cofactor expansion
5. Matrix Inverse: A⁻¹ = (1/det(A)) × adj(A)
6. Transpose: Aᵀ[i,j] = A[j,i]
Common Matrix Types:
- Square Matrix: Same number of rows and columns
- Diagonal Matrix: Non-zero elements only on main diagonal
- Identity Matrix: Diagonal matrix with all 1's on diagonal
- Symmetric Matrix: A = Aᵀ (equal to its transpose)
- Orthogonal Matrix: A × Aᵀ = I (columns are orthonormal)
Dimension Requirements:
- Addition/Subtraction: Matrices must have identical dimensions
- Multiplication: A(m×n) × B(n×p) = C(m×p)
- Determinant: Only defined for square matrices
- Inverse: Only square matrices with non-zero determinant
Tip: Use the step-by-step solutions to understand the underlying mathematical processes.