Matrices

Understanding Matrices

A matrix is a rectangular grid of numbers used to represent linear transformations, systems of equations, and structured data. Discrete mathematics uses matrix operations to solve systems efficiently, represent graphs (adjacency matrices), and model transformations — skills that carry directly into computer graphics, machine learning, and network analysis.

Key Definitions & Formulas

• Matrix addition: add corresponding entries; requires matrices of the same dimensions.
• Matrix multiplication: the entry in row i, column j of A×B is the dot product of row i of A and column j of B.
• Determinant: a single number summarizing a square matrix, non-zero exactly when the matrix is invertible.
• Inverse (A⁻¹): the matrix such that A × A⁻¹ = the identity matrix, used to "undo" a transformation.
• Transpose (Aᵀ): flips a matrix over its diagonal, swapping rows and columns.

Worked Example

Multiplying a 2×2 matrix [[1,2],[3,4]] by [[5,6],[7,8]] gives [[1×5+2×7, 1×6+2×8], [3×5+4×7, 3×6+4×8]] = [[19,22],[43,50]] — each output entry is the dot product of a row from the first matrix and a column from the second.

Where This Is Used

• Computer graphics transformations (rotation, scaling, projection).
• Representing and analyzing graphs via adjacency matrices.
• Solving systems of linear equations in engineering and economics.
• Machine learning, where data and model weights are stored and manipulated as matrices.