**Definition:** A matrix A over a field K or, simply, a matrix A is a regular array of scalars usually presented in the following form

The rows of such a matrix A are the m horizontal lists of scalars:

## and the terms of A are the n verticals lists of scalars:

**Matrix addition and scalar multiplication:**

be two matrices with the same size, say m×n matrices. The sum of A and B, written A+B, is the matrix obtained by adding corresponding elements from A and B. That is

The product of the matrix A by a scalar k, written k.A or simply kA, is the matrix obtained by multiplying each element of A by k. That is,

**Matrix multiplication:**

The product of matrix A and B written AB, is some what complicated. For this reason, we first begin with a special case.

The product AB of a row matrix

with the same number of elements is defined to be the scalar obtained by multiplying corresponding entries and adding; that is

**Transpose of a matrix:** The transpose of a matrix A, written AT, is the matrix obtained by writing the columns A, in order, as rows.

For example,

In other words, if A = [a_{ij}] is an m×n matrix, then A^{T}=[b_{ij}] is the n×m matrix where b_{ij}=a_{ji}

**Square matrices**

A square matrix is a matrix with the same number of rows as columns. An n×n square matrix is said to be of order n and is sometimes called an n – square matrix.

**Diagonal and Trace**

Let A = [a_{ij}] be an square matrix. The diagonal or main diagonal of A consists of the elements with the same subscripts, that is,

The trace of A, written tr(A), is the sum of the diagonal elements. Namely,

**Identity matrix, Scalar matrices**

The n-square identity or unit matrix, denoted by In, or simply I, is the n- square matrix with I’s on the diagonal and 0’s elsewhere. The identity matrix I is similar to the scalar 1 in that, for any n- square matrix.