N×NMatrix

An n×n matrix is a square array of numbers or symbols with n rows and n columns. It is usually denoted A ∈ F^{n×n}, where F is a field such as the real numbers R or complex numbers C. The entry in the i-th row and j-th column is written a_ij. Square matrices arise naturally in linear algebra as representations of linear transformations on n-dimensional space.

For example, a 2×2 matrix is [ [a11, a12], [a21, a22] ]. A 3×3 matrix has nine entries arranged in three rows and three columns.

Operations: matrices can be added elementwise and multiplied by scalars. If A ∈ F^{n×n} and B ∈ F^{n×n}, then AB is defined and is also n×n. Transpose A^T swaps rows and columns. In general, matrix multiplication corresponds to composing linear transformations, and the identity matrix I_n acts as the neutral element for multiplication.

Determinant and invertibility: det(A) is defined for square matrices; det ≠ 0 implies A is invertible and there exists A^-1 with AA^-1 = I_n. The determinant provides information about area, volume, solvability of linear systems, and sensitivity to changes.

Special matrices and properties: the identity I_n, diagonal matrices, symmetric matrices (A^T = A), and skew-symmetric matrices (A^T = −A). Real symmetric matrices have real eigenvalues and, with appropriate conditions, an orthogonal eigenbasis; positive definite and semidefinite matrices are central in optimization and statistics.

Applications: solving systems Ax = b, representing linear transformations, computer graphics, and numerical methods. Common algorithms include Gaussian elimination, LU decomposition, and eigenvalue decompositions, which enable analysis and computation with n×n matrices across disciplines.