By an m x n matrix we mean an array of m x n elements, for example, ak (i = 1, ..., m; k = 1, ..., n) arranged in a rectangular form:

The element akk is called the (i, k)-element (entry or component). The notation for matrix is often abbreviated by writing A = (aik).

Each horizontal n-tuple in an m x n matrix is called a row of the matrix, and each vertical m-tuple is called a column of the matrix. The m x n matrix B, where bk = aki is called the transposed matrix of A and denoted as AT

An n x n matrix is called a square matrix of order n. A square matrix is called diagonal matrix if all its components are zero except for diagonal components a;;. If the components a;; of a diagonal matrix are all equal to 1, it is called a unit matrix or identity matrix and denoted by I.

A 1 x n matrix is called a row vector of dimension n, and m x 1 matrix is called a column vector of dimension m.

Operation on Matrices

If both A and B are m x n matrices, then their sum is defined by A + B = (aik + bik). If A is an m x n matrix and B is an n x m matrix, their product is defined by

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