Matrices: Foundations, Operations, and Applications

Matrices are fundamental mathematical objects used to represent and manipulate data in a structured way. In its simplest form, a matrix is a rectangular array of numbers arranged in rows and columns. Matrices are essential in linear algebra, and they appear in nearly every area of mathematics and its applications, including computer science, physics, and statistics.

1. Definition of a Matrix

A matrix of size is an array of numbers with rows and columns:


Each is an element of the matrix, called the -th entry.

2. Types of Matrices

• Row matrix: One row, multiple columns.
• Column matrix: One column, multiple rows.
• Square matrix: Same number of rows and columns.
• Zero matrix: All entries are zero.
• Identity matrix : A square matrix with on the diagonal and elsewhere.
• Diagonal matrix: Nonzero entries only on the diagonal.
• Symmetric matrix: A matrix such that .

3. Matrix Operations

• Addition: is defined if and have the same dimensions:
  
• Scalar Multiplication: scales each entry by :
  
• Matrix Multiplication: If is and is , then is :
  
• Transpose: The transpose switches rows and columns:
  

4. Determinants

For a square matrix , the determinant (or ) is a scalar value that reflects various properties such as invertibility. For example:

For a matrix:

A square matrix is invertible if and only if .

5. Inverse of a Matrix

If is an invertible matrix, then its inverse satisfies:


The inverse can be computed via row reduction, the adjugate matrix formula, or using Gauss–Jordan elimination.

6. Solving Linear Systems with Matrices

A system of linear equations can be written in matrix form as:


Where:

• is the coefficient matrix,
• is the column vector of variables,
• is the column vector of constants.

If is invertible, the solution is:


Alternatively, methods such as Gaussian elimination or LU decomposition can be used.

7. Eigenvalues and Eigenvectors

Let be a square matrix. A nonzero vector is an eigenvector of if:


for some scalar , called the eigenvalue associated with . The eigenvalues of are roots of the characteristic polynomial:


8. Applications of Matrices

• Geometry: Matrices represent rotations, reflections, and projections.
• Computer Graphics: Transformations of objects in 2D/3D.
• Physics: Quantum mechanics, relativity, and systems of equations.
• Statistics: Covariance matrices and multivariate data analysis.
• Markov Chains: State transition matrices describe probability evolution.
• Network Theory: Adjacency matrices represent graphs.

9. Matrix Representations in Graph Theory

• Adjacency Matrix: if there is an edge from vertex to .
• Incidence Matrix: Describes relationships between vertices and edges.

10. References

• Wikipedia: Matrix (mathematics)
• Gilbert Strang – Linear Algebra and Its Applications
• AoPS Wiki: Matrix
• Axler – Linear Algebra Done Right