Matrices

DEFINITION

A rectangular array of symbols (which could be real or complex numbers) along rows and columns is called a matrix.
Thus a system of m x 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 × n matrix).

$ \large \left[ \begin{array}{ccc} a_{11} & a_{12} ….. & a_{1n}\\ a_{21} & a_{22}  …..& a_{2 n}  \\ ……\\……\\ a_{m1} & a_{m2} ….. & a_{mn} \end{array} \right] $ is a matrix of order m × n

In a compact form the above matrix is represented by A = [aij], 1 ≤ i ≤ m, 1 ≤j ≤ n or simply [aij]m x n
The numbers a11, a12, … etc of this rectangular array are called the elements of the matrix. The element aij belongs to the ith row and jth column and is called the (i, j)th element of a matrix.

Equal Matrices:

Two 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 Matrix:
A 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.

$ \large \left[ \begin{array}{c} 2 \\ 3 \\ 5 \end{array} \right] $

Square Matrix

An m × n matrix A is said to be a square matrix if m = n i.e. number of rows = number of columns.
For example: $ \large A = \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{array} \right] $ is a square matrix of order 3 × 3

Note:
⋄ 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.

Trace of a Matrix

The sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A i.e. tr(A).  Thus if A = [aij]n×n , Then

$\large tr(A) = \Sigma_{i=1}^{n} a_{ii} $

$\large = a_{11} + a_{22} + a_{33} + ….. + a_{nn}$

Properties of Trace of a Matrix:

Let A = [aij]nn and B = [bij]nn and λ be a scalar

(i) tr(λA) = λ tr(A)

(ii) tr(A + B) = tr(A) + tr(B)

(iii) tr(AB) = tr(BA)

Diagonal Matrix:

A square matrix all of whose elements except those in the leading diagonal, are zero is called a diagonal matrix. For a square matrix A = [aij]n×n to be a diagonal matrix, aij = 0, whenever i ≠ j.

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

For example:

$ \large A = \left[ \begin{array}{ccc} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -1 \end{array} \right] $ is a diagonal matrix of order 3 × 3

Scalar Matrix:

A diagonal matrix whose all the leading diagonal elements are equal is called a scalar matrix. For a square matrix A = [aij]n×n to be a scalar matrix

$ \large a_{ij} = \left\{\begin{array}{ll} 0  ,  &  i \ne j \\ m ,  &  i = j \end{array}  \right.  $

Where m ≠ 0

For example:

$ \large A = \left[ \begin{array}{ccc} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{array} \right] $ is a scalar matrix .

Unit Matrix or Identity Matrix:

A diagonal matrix of order n which has unity for all its diagonal elements, is called a unit matrix of order n and is denoted by In

Thus a square matrix A = [aij]n×n is a unit matrix if

$ \large a_{ij} = \left\{\begin{array}{ll} 1  ,  &  i =  j \\ 0 ,  &  i \ne  j \end{array}  \right.  $

For example:

$ \large A = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] $

Triangular Matrix

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

Given a square matrix A = [aij]nxn

For upper triangular matrix, aij = 0, i > j

and for lower triangular matrix, aij = 0, i < j

Note:
⋄ Diagonal matrix is both upper and lower triangular.

⋄ A triangular matrix A = [aij]nxn is called strictly triangular if aii = 0 for 1 ≤ i ≤ n.

For example:

$ \large \left[ \begin{array}{ccc} a & h & g \\ 0 & b & f \\ 0 & 0 & c \end{array} \right]  \; and \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 1 & -5 & 4 \end{array} \right]$  are respectively upper and lower triangular matrices.

Null Matrix:

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

For A = [aij] to be null matrix, aij = 0 ∀ i, j

For example: $\large \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] $
is a zero matrix

Transpose of a Matrix:

The matrix obtained from any given matrix A, by interchanging rows and columns, is called the transpose of A and is denoted by A’.
If A = [aij]mxn and A’ = [bij]nxm, then bij = aji, ∀ i, j

Properties of Transposes:

(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

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