Matriz

Matriz is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. In mathematics, a matrix is denoted by a capital letter A, with entries aij at the i-th row and j-th column. Matrices can be real or complex and have dimensions m×n.

Operations on matrices include addition and subtraction, performed element-wise when the matrices share the same dimensions; scalar multiplication, which multiplies every entry by a scalar. Matrix multiplication combines an m×p matrix A with a p×n matrix B to yield an m×n matrix C = AB; unlike ordinary multiplication, it is not generally commutative.

Key concepts include the transpose, denoted A^T, which flips rows and columns; symmetry, where A^T = A; and invertibility for square matrices, with an inverse A^{-1} satisfying AA^{-1} = A^{-1}A = I. The identity matrix I_n serves as the multiplicative identity. The rank of a matrix is the maximum number of linearly independent rows or columns, obtainable via row reduction. Common decompositions such as LU, QR, and eigendecomposition (or diagonalization) aid in solving linear systems and analyzing linear operators. Special matrices include the zero matrix, diagonal and triangular matrices, and orthogonal matrices where A^T A = I.

Applications of matrices are broad: they describe linear transformations, systems of equations, and data in science and engineering. They are central in computer graphics, physics simulations, statistics (e.g., covariance matrices), and machine learning.

In Portuguese and Spanish, matriz denotes the same mathematical concept. The term also appears in other domains: in biology as extracellular matrix and in business to refer to the parent company (empresa matriz).