Core Matrix Operations:

Key Formulas:

Matrix Addition: C[i,j] = A[i,j] + B[i,j]
Matrix Multiplication: C[i,j] = Σ A[i,k] × B[k,j]
Determinant (2×2): det = a₁₁×a₂₂ - a₁₂×a₂₁
Determinant (3×3): Sarrus rule or cofactor expansion
Matrix Inverse: A⁻¹ = (1/det(A)) × adj(A)
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.

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