Basic Matrix Operations
Fundamental arithmetic for matrices.
Addition & Subtraction
Performed element-wise. Matrices must have the exact same dimensions.
[A] + [B] = [Aᵢⱼ + Bᵢⱼ]
Scalar Multiplication
Multiply every single element in the matrix by a constant number (scalar).
k · [A] = [k · Aᵢⱼ]
Transpose
Flip the matrix over its main diagonal. Rows become columns, and columns become rows.
(Aᵀ)ᵢⱼ = Aⱼᵢ
Trace
The sum of the elements on the main diagonal (top-left to bottom-right). Only for square matrices.
tr(A) = Σ Aᵢᵢ
Determinant
A scalar value describing the scaling factor of the linear transformation. If zero, the matrix is not invertible.
det(A) or |A|
Adjoint (Adjugate)
The transpose of the cofactor matrix. Crucial for finding the inverse manually.
adj(A) = Cᵀ
Inverse
The matrix that yields the Identity matrix when multiplied with the original. Only exists if det(A) ≠ 0.
A⁻¹ = (1/|A|) · adj(A)