Perform addition, subtraction, matrix multiplication, determinants, inverses, transposes, powers, and rank calculations with exact rational fractions.
In linear algebra, a matrix is a rectangular array of numbers, symbols, or expressions arranged in horizontal rows and vertical columns.
The dimensions of a matrix are written as m × n, where m represents the number of rows and n represents the number of columns. An individual element at row i and column j is denoted as ai,j.
Requires both matrices to have identical dimensions (m_A = m_B and n_A = n_B). Computed by adding or subtracting corresponding elements:
Requires the number of columns in A to match the number of rows in B (n_A = m_B). The product matrix has dimensions m_A × n_B:
⚠️ Non-Commutative: In general, AB ≠ BA.
A scalar value computed only for square matrices (n × n). For a 2×2 matrix:
The matrix satisfying A · A⁻¹ = I. Exists if and only if the matrix is square and det(A) ≠ 0 (non-singular):
3D game engines and shaders use 4×4 transformation matrices to rotate, translate, scale, and project 3D polygonal vertices onto 2D screens.
Engineering circuits, structural load distributions, and economic models formulate large linear systems as Ax = b, solved via matrix inversion or LU decomposition.
Artificial neural networks compute forward passes by multiplying input vectors by layer weight matrices: y = σ(W · x + b).
Because matrix multiplication represents the composition of linear transformations. The order in which transformations are applied matters (e.g. rotating an object then translating it yields a different position than translating first then rotating).
A matrix with det = 0 is called a singular matrix or degenerate matrix. Geometrically, it collapses space into a lower dimension (e.g. 2D plane into a 1D line), meaning the transformation cannot be reversed, so no inverse matrix exists.
The identity matrix (I) is a square matrix with 1s along the main diagonal and 0s everywhere else. Multiplying any compatible matrix by the identity matrix leaves it unchanged: A · I = A.