n×nmatriisien

An n×n matrix, often referred to as a square matrix, is a fundamental object in linear algebra. It is a two-dimensional array of numbers, symbols, or expressions arranged in rows and columns, where the number of rows is equal to the number of columns. This equality defines its "square" nature, with 'n' representing the number of rows and also the number of columns.

Square matrices play a crucial role in representing linear transformations between vector spaces of the same dimension. For instance, a linear transformation from an n-dimensional space to itself can be uniquely described by an n×n matrix. Operations such as matrix addition, scalar multiplication, and matrix multiplication are defined for square matrices. Matrix multiplication, in particular, is associative and has special properties when dealing with square matrices, leading to concepts like the identity matrix and the inverse matrix.

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides significant information about the matrix, including whether it is invertible (non-singular). A square matrix has an inverse if and only if its determinant is non-zero. The trace of a square matrix is the sum of the elements on its main diagonal. Eigenvalues and eigenvectors are also intrinsically linked to square matrices and are vital in understanding the behavior of linear transformations and solving systems of differential equations.