A matrix is a rectangular array of constants or variables that can be operated on mathematically. The numbers or variables in a matrix are organized in horizontal rows and vertical columns, usually enclosed by brackets.

A matrix of m rows and n columns is called a matrix with dimensions m × n (read "m by n").

In a matrix, numbers or data are organized so that each position in the matrix has a purpose.  Each value in the matrix is called an element.

A matrix is usually names using an uppercase letter, but can also be named by using the matrix dimensions with the letter name.  The dimensions tell how many rows and columns, in that order, are in the matrix.  Some examples follow.

3 × 3 matrix	1 × 3 matrix	2 × 3 matrix

Since the numbers that make up a matrix are called elements, we know that the elements (or entries) used to make up matrix M above are all numbers, but the matrix itself is not a number, any more than a multiplication table is a number. Various operations can be defined on matrices, including addition, subtraction, and multiplication. 

In order to distinguish real numbers from a matrix, real numbers are called scalars.  For example, you can multiply any matrix by a constant called a scalar.  This is called scalar multiplication.  When scalar multiplication is performed, each element is multiplied by that constant, and a new matrix if formed.

example:

Two matrices are considered to be equal if they have the same dimensions, and if each element of one matrix is equal to the corresponding element of the other matrix.

Many problems can be solved using a method sometimes referred to as matrix logic.  When you use matrix logic, you create a matrix that helps you organize all the information in the problem.  By using the matrix, you can eliminate one possibility after another until you eventually arrive at a solution.

The Favorite Sports Four students named, Brian, Frank, Hector, and Steven each has a favorite sport.  These sports are basketball, fishing, hiking, and soccer, but not necessarily respectively.  The following information is known about the students and their favorite sports:   No student's name begins with the same letter as his or her favorite sport.     Frank and Hector like team sports.   Brian spent last summer vacationing at a lake, where he spent most of   his time fishing.   If Steven likes hiking.  Frank doesn't like basketball.   What is the favorite sport of each student?