A matrix (plural : matrices) is a rectangular arrangement of numbers in rows and columns. It is written in between a pair of rectangular brackets. The numbers belonging to a matrix are known as its elements. Matrices are represented by capital Roman letters. For example, consider the following matrix .
It has 2 rows and 3 columns. Hence it is a matrix.
If a matrix has rows and columns, it is said to be of the order by . So, is of the order 2 by 3 and has 6 elements.
Representation of an Element
corresponds to that element of a matrix, which is present in the th row and th column of it. So, for matrix ,
Transpose of a matrix
A matrix obtained by interchanging the rows and columns of a matrix is known as the transpose of the matrix. So,
Equality of 2 matrices
2 matrices A and B are equal iff (if and only if)
i) their order is same and
ii) in and in ,
A row matrix has only 1 row and more than 1 column. e.g.
A column matrix has only 1 column and more than 1 row. e.g.
A null matrix is the one, whose all elements are . e.g.
A square matrix has equal number of rows and columns. So, the total number of elements in a square matrix is always a perfect square. e.g.
So, has 4 rows as well as 4 columns, so 16 elements.
The elements of a square matrix, for which are known as the diagonal elements. Thus, and are diagonal elements of .
A square matrix, whose all elements are zero, except the diagonal elements, is known as the diagonal matrix.e.g.
An identity matrix is a square matrix, whose diagonal elements are equal to 1 and all other elements are 0. This matrix is of special importance in the matrix theory. Similar to the number in the theory of numbers, it acts as a unit entity. It is generally denoted by . So, for any square matrix , . Note that and must have the same order.
Upper- and Lower-Triangular Matrices
An upper-triangular matrix is the one, whose elements below the diagonal are . Similarly, a lower-triangular matrix is the one, whose elements above the diagonal are .