MxN

In linear algebra, an M-by-N matrix (written M×N or M by N) is a rectangular array of numbers with M rows and N columns. It is commonly denoted A ∈ F^{M×N}, where F is a field such as the real numbers R or complex numbers C. The dimensions M and N are counts of rows and columns, respectively.

An M×N matrix can represent a linear transformation from an N-dimensional space to an M-dimensional space. If x ∈ F^N is a column vector, then the product y = A x lies in F^M. The columns of A correspond to the images of the standard basis vectors in F^N, and the rows constrain linear relations among the elements of x.

Examples help illustrate the shape. A 2×3 matrix has 2 rows and 3 columns, such as A = [ [1, 2, 3], [4, 5, 6] ]. A 3×2 matrix has 3 rows and 2 columns, such as B = [ [7, 8], [9, 10], [11, 12] ].

Operations and properties depend on the dimensions. Matrix addition and subtraction require the same dimensions (MxN). Multiplication is defined for A (M×N) and B (N×P), yielding a product C with dimensions M×P. The transpose A^T has dimensions N×M. The rank of an M×N matrix is the maximum number of linearly independent rows or columns and satisfies rank ≤ min(M, N). If M = N, the matrix is square; certain square matrices are invertible and have determinants and eigenvalues.

M×N matrices are used to organize data, represent linear systems, and model transformations in science and engineering. In many contexts, the term “M by N” is read aloud as “M by N.”