Definition of a Matrix

If m and n are positive integers, then m x n matrix is a rectangular array,

          

     a11 a21  am1 a12 a22  am2     a1n a2n  amn              x1 x2  xn        =      b1 b2  bm        

x is a column vector with n entries, and b is a column vector with m entries.

An m x n matrix (read "m" by "n") has m rows and n columns.

The plural of matrix is matrices. If each entry of a matrix is a real number, then the matrix is called a real matrix. 

A matrix with m rows and n columns (an m x n matrix) is said to be of size m x n. If m = n, the matrix is called a square of order n. For a square matrix, the entries a11, a22, are called  the main diagonal entries.

Examples of Matrices:

Elementary Row Operations
1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.

Two matrices are said to be row-equivalent if one can be obtained from the other by a finite sequence of elementary row operations.

A matrix is said to be in row-echelon form if:
1. All rows consisting entirely of zeros occur at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading coefficient).
2. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.
Example:
A1=   1 0 0 0 1 0 0 0 1         2 3 −4    

A2=   1 0 0 0 1 0 −3 2 0         −5 4 0     

A3=   1 0 0 0 1 0 0 0 0         3 2 1