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Applications of matrices are found in most scientific fields. Infinite matrices occur in planetary theory and in atomic theory. The size of a matrix is defined by the number of rows and columns that it contains. The specifics of symbolic matrix notation vary widely, with some prevailing trends.

In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. This article follows the more common convention in mathematical writing where enumeration starts from 1. An asterisk is occasionally used to refer to whole rows or columns in a matrix. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain.

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Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram. These vectors define the vertices of the unit square. The blue original is mapped to the green grid and shapes.

Rotation by pi over 6. The last equality follows from the above-mentioned associativity of matrix multiplication. Any two square matrices of the same order can be added and multiplied. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.