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In mathematics, a matrix is a two dimensional rectangular array of numbers. Matrices play an extremely important role in many areas of mathematics, particularly linear algebra. They are a compact way of representing any linear dependence of some numbers on some other numbers.

Matrix of elements [entries in the two dimensional array] that may be real, integers or complex.

Matrix as a stalking horse, under choice of vector bases, for a linear operator.

Matrices as abbreviation for sets of simultaneous linear equations.

Matrix equality, addition, subtraction, multiplication, division [multiplication by the inverse].

Matrix conformability for multiplication.

Matrix multiplication is generally non-commutative - changing the order in which the matrices are multiplied gives a different product matrix.

Matrix inversion of non-singular matrices.

Determinants.

Cramer's rule.

Gaussian elimination for systems of linear simultaneous equations.

Gauss-Jordan solution for a system of linear simultaneous equations.

Symmetric matrices.

Orthogonal matrices.

Complex matrices; Hermitian, unitary.