Matrices Learning

A set of  mn numbers (real or imaginary) arranged in the form of a rectangular array of m rows and n columns is called an m × n matrix (to be real as ‘m by n’ matrix)

An m × n matrix is usually written as

matrix(A)

In compact form the above matrix is represented by A = [aij]m × n or A = [aij].

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The numbers a11, a12, .. etc. are known as the elements of the matrix A. The elements aij belongs to ith row and jth column and is called the (I, j)th element of the matrix A = [aij]. Thus, in the element aijthe first subvscript I always denotes the number of rows and the second subscript j, number of columns in which the elements occurs. Example for learning matrices is explained below.

(i)          matrix(B)  is matrix having 2 rows and 3 columns and so kit is a matrix of order 2 × 3 such that a11 = 2, a12 = 1, a13 = – 1, a21 = 1, a22 = 3, a23 = 2.

matrices learning- types

Row Matrix:- A matrix having only one row is called a row-matrix or a row- vector

Column Matrix:- Amatrix having only one column is called acolumn matrix or a column-vector.

Square Matrix:- Amatrix in which the number of rows is equal to the number of columns, say n, is called a square matrix of order n.

A square matrix of order n is also called a n-rowed square matrix. The elements aij of a square matrix A = [aij]n × n for which I = j, i.e., the elements a11, a22, …., ann are called the diagonal elements and the line along which they lie is called the principal diagonal or leading diagonal of the matrix.

Diagonal Matrix:- A square matrix A = [aij]n × n is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero i.e.

aij = 0 for all i = j

A diagonal matrix of order n × n having d1, d2, ….dnas diagonal elements is denoted by diag [d1, d2, …dn].

matrices learning- important types

Scalar Matrix:- A square matrix A = [aij]n × n is called a scalar matrix if

(i)            aij = 0 for all i ≠ j, and

(ii)          aij = C for all i, where C ≠ 0

in other words, a diagonal matrix in which all the diagonal elements are equal is called the scalar matrix

Identity Or Unit Matrix:- A square matrix A = [aij]n × n is called an identity or unit matrix if

(i)            aij = 0 for all i ≠ j and

(ii)          aij = 1 for all i

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In other words, a square matrix each of whose diagonal elements is unity and each of whose non-diagonal elements is equal to zero is called and identity or unit matrix.

The identity matrix of order n is denoted by In.

Null Matrix:- A matrix whose all elements are zero is called a null matrix or a zero matrix.