In
mathematics, a
matrix (plural
matrices) is a
rectangular of
numbers,
symbols, or
expressions, arranged in
s and
s. The dimensions of the matrix below are
2 ×
3 (read "two by three"), because there are two rows and three columns.
The individual items in a matrix are called its
elements or
entries. Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be
added or subtracted element by element. The rule for
matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a
scalar from its associated
field. A major application of matrices is to represent
linear transformations, that is, generalizations of
linear functions such as . For example, the
rotation of
vectors in three
dimensional space is a linear transformation which can be represented by a
rotation matrix R: if
v is a
column vector (a matrix with only one column) describing the
position of a point in space, the product
Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the
composition of two linear transformations. Another application of matrices is in the solution of
systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its
determinant. For example, a square matrix
has an inverse if and only if its determinant is not
zero. Insight into the
geometry of a linear transformation is obtainable (along with other information) from the matrix's
eigenvalues and eigenvectors.