Definition of Matrices
Matrices (or matrix in singular), is a compact notation method for element arrays. Often matrices are used with equations systems and vectors. The elements can then be easily treated.
Consider the following system of algebraic equations:
In matrix form it will be written as:
In general, an equation system:
Can be written in matrix form:
And in a more compact notation:
Note that the matrix is represented by uppercase bold letters and the elements by lowercase letters.
In the previous example, the matrix A of coefficients a, has m rows and n columns. The matrix X of variables x has n rows and only one column. The matrix Y of y has m rows and one column.
When we refer to an element in a matrix that has two dimensions, we first name the number of its row and then the number of its column. As we can see in the matrix A of coefficients a, in the first row, the first number in all the coefficients is number one, and changes only the second number, denoting the column of each element. In the second row, the first number is the number two and the second number, again changes to denote each element’s column.
In the case that a matrix has the same number of rows and columns, it is called square matrix. For example the following matrix is a 3X3 square matrix:
Diagonal is the matrix that has all its elements equal to zero, except those in the main diagonal:
Unit or Identity Matrix
A square matrix that has the elements of its main diagonal equal to one and all the others equal to zero is called unit matrix. A unit matrix of n rows and columns is denoted as In. A unit matrix of three rows and columns is: