## Matrices: Definition and Classification

DEFINITIONA rectangular array of symbols (which could be real or complex numbers) along rows and columns is called a matrix.

Thus a system m × n symbols arranged in a rectangular formation along m rows and n columns and bounded by the brackets [.] is called an m by n matrix (which is written as m x n matrix)

i.e. A = is a matrix of order m × n.

In a compact form the above matrix is represented by A = [a

_{ij}], 1 < i < m,

1 < j < n, where is, j ε i or simply [a_{ij}] m × n.The numbers a

_{11}, a_{12}, ... etc of this rectangular array are called the elements of the matrix. The element a_{ij}belongs to the i^{th}row and the j^{th}column and is called the (i, j)^{th}element of the matrix.

Equal MatricesTwo matrices are said to be equal if they have the same order and each element of one is equal to the corresponding element of the other.

CLASSIFICATION OF MATRICES

Row MatrixA matrix having a single row is called a row matrix. e.g. [1 3, 5, 7]

Column Matrix

_{A matrix having a single column is called a column matrix. e.g.}.

Square MatrixAn m x n matrix A is said to be a square matrix if m = n i.e. number of rows = number of columns.

For example:A = is a square matrix of order 3 × 3.

Note:In a square matrix the diagonal from left hand side upper corner to right hand side lower corner is known as leading diagonal or principal diagonal. In the above example square matrix containing the elements 1, 3, 5 is called the leading or principal diagonal.

Traces of a MatrixThe sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A i.e. t

_{r}(A). Thus if A = [a_{ij}]_{n×n},then t

_{r}(A) = ∑^{n}_{i=1 }a_{ii}= a_{11}+ a_{22}+......+ a_{nn}.

Diagonal MatrixA square matrix all of whose elements except those in the leading diagonal, are zero is called a diagonal matrix. For a square matrix A = [a

_{ij}]_{n×n}to be a diagonal matrix, a_{ij}= 0, whenever i ≠ j.

For example:A = is a diagonal matrix of order 3 × 3.

Note:Here A can also be represented as diag (3,5 -1).

Scalar MatrixA diagonal matrix whose all the elements are equal is called a scalar matrix.

For a square matrix A = [a

_{ij}]_{n×n}to be a scalar matrix, a_{ij}=, where m ≠ 0.For example: A = is a scalar matrix.

Unit Matrix or Identity MatrixA diagonal matrix of order n which has unity for all its elements, is called a unit matrix of order n and is denoted by I

_{n}.Thus a square matrix A = [a

_{ij}]_{n×n}is a unit matrix if a_{ij}=.

For example :I

_{3}=.

Triangular MatrixA square matrix in which all the elements below the principal diagonal are zero is called Upper Triangular matrix and a square matrix in which all the elements above the principal diagonal are zero is called Lower Triangular matrix.

Given a square matrix A = [a

_{ij}]_{n×n},for upper triangular matrix, a

_{ij}= 0, i > jand for lower triangular matrix, a

_{ij}= 0, i < j.

Note :

* Diagonal matrix is both upper and lower triangular

* A triangular matrix A = [a_{ij}]_{n×n}is called strictly triangular if a_{ii}= 0 for 1 < i < n.

For example :are respectively upper and lower triangular matrices.

Null MatrixIf all the elements of a matrix (square or rectangular) are zero, it is called a null or zero matrix.

For A = [a

_{ij}] to be null matrix, a_{ij}= 0 i, j.

For example:is a zero matrix.

Transpose of a MatrixThe matrix obtained from any given matrix A, by interchanging its rows and columns, is called the transpose of A and is denoted by A'.

If A = [a

_{ij}]_{m×n}and A' = [b_{ij}]_{n×m}then b_{ij}= a_{ij}, i, j.

For example:If A = , then A' =

Properties of Transpose(i) (A')' = A

(ii) (A + B)' = A' + B', A and B being conformable matrices

(iii) (αA)' = αA', α being scalar

(iv) (AB)' = B'A', A and B being conformable for multiplication

Conjugate of a MatrixThe matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex numbers is called conjugate of A and is denoted by .

For example:A = . Then,

Properties of Conjugate

Transpose conjugate of a MatrixThe transpose of the conjugate of a matrix is called transposed conjugate of A and is denoted by A

^{q}. The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e.= A

^{Θ}.If A = [a

_{ij}]_{m×n}, then A^{Θ}= [b_{ij}]_{n×m}where b_{ji}=i.e. the (j, i)

^{th}element of A^{Θ}= the conjugate of (i, j)^{th}element of A.

For Example:If A = then A

^{Θ}= .

Properties of Transpose conjugate(i) (A

^{Θ})^{Θ}= A(ii) (A + B)

^{Θ}= A^{Θ}+ B^{Θ}(iii) (kA)

^{Θ}= A^{Θ}, k being any number(iv) (AB)

^{Θ}= B^{Θ}A^{Θ }The platform at askIITians is the only group of IITians which provide you free online courses and a good professional advice for IIT JEE, AIEEE and other Engineering Examination preparation. You can also visit askIITians.com to study topics pertaining to the IIT JEE and AIEEE syllabus for free.