TOPICS

Introduction

Matrices

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

The horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix.

A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix). In general form, an m x n matrix has the rectangular array : or A = [aᵢⱼ]m × n, 1≤ i ≤ m, 1≤ j ≤ n i, j ∈ N Thus the ith row consists of the elements aᵢ₁, aᵢ₂, aᵢ₃,..., aᵢn, while the jth column consists of the elements a₁ⱼ, a₂ⱼ, a₃ⱼ,..., amⱼ,

There have many types of matrix following below :

(i) Row matrix

(ii) Column matrix

(iii) Square matrix

(iv) Diagonal matrix

(v) Scalar matrix

(vi) Identity matrix

(vii) Zero matrix

A matrix is said to be a column matrix if it has only one column. For Example, In general, B = [bᵢⱼ] 1 × n is a row matrix of order 1 × n.

A matrix is said to be a column matrix if it has only one column. For Example, In general, A = [aᵢⱼ] m × 1 is a column matrix of order m × 1.

A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For Example, In general, A = [aᵢⱼ] m × m is a square matrix of order m.

A square matrix B = [bᵢⱼ] m × m is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = [bᵢⱼ] m × m is said to be a diagonal matrix if bᵢⱼ = 0, when i ≠ j. For Example,

A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [bᵢⱼ] n × n is said to be a scalar matrix if bᵢⱼ = 0, when i ≠ j bᵢⱼ = k, when i = j, for some constant k. For Example,

A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. For Example,

A matrix is said to be zero matrix or null matrix if all its elements are zero. We denote zero matrix by O. We denote the identity matrix of order n by In. When order is clear from the context, we simply write it as I. For Example,

Two matrices A = [aij] and B = [bij] are said to be equal if

(i) they are of the same order

(ii) each element of A is equal to the corresponding element of B, that is aᵢⱼ = bᵢⱼ for
all i and j. For example,

There have some operations on matrice following bwlow :

(i) Addition of matrices

(ii) Difference of matrices

(iii) Scalar multiplication of matrices

(iv) Negative of a matrices

(v) Multiplication of matices

The sum of two or more matrices is called addition of matrices. For Example, In general, if A = [aᵢⱼ] and B = [bᵢⱼ] are two matrices of the same order, say m × n. Then, the sum of the two matrices A and B is defined as a matrix C = [cᵢⱼ]m × n, where cᵢⱼ = aᵢⱼ + bᵢⱼ, for all possible values of i and j.

The diffrence between two matrices is called difference of matrices. For Example, In general, if A = [aᵢⱼ] and B = [bᵢⱼ] are two matrices of the same order, say m × n. Then, the difference of the two matrices A and B is defined as a matrix C = [cᵢⱼ]m × n, where cᵢⱼ = aᵢⱼ - bᵢⱼ, for all possible values of i and j.

A matrix in which multiply by any scalar value is called scalar multiplication of matrices. For Example,
In general, we may define multiplication of a matrix by a scalar as follows: if
A = [aᵢⱼ] m × n is a matrix and k is a scalar, then kA is another matrix which is obtained
by multiplying each element of A by the scalar k.

In other words, kA = k [aᵢⱼ] m × n = [k (aᵢⱼ)] m × n, that is, (i, j)th element of kA is kaᵢⱼ
for all possible values of i and j.

The negative of a matrix is denoted by –A. We define –A = (– 1) A. For example,

The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = [aᵢⱼ] be an m × n matrix and B = [bjk] be an n × p matrix. For example,

If A = [aᵢⱼ] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′ or (AT). In other words, if A = [aᵢⱼ]m × n, then A′ = [aⱼᵢ]n × m. For example,

A square matrix A = [aᵢⱼ] is said to be symmetric if A′ = A, that is, [aᵢⱼ] = [aⱼᵢ] for all possible values of i and j. For Example,

A square matrix A = [aᵢⱼ] is said to be skew symmetric matrix if A′ = – A, that is, [aⱼᵢ] = -[aᵢⱼ] for all possible values of i and j. For Example, This means that all the diagonal elements of a skew symmetric matrix are zero.

There are six operations (transformations) on a matrix, three of which are due to rows
and three due to columns, which are known as elementary operations or
transformations.
Rule 1. The interchange of any two rows or two columns.

Rule 2. The multiplication of the elements of any row or column by a non zero number.

Rule 3. The addition to the elements of any row or column, the corresponding
elements of any other row or column multiplied by any non zero number.

If A is a square matrix of order m, and if there exists another square
matrix B of the same order m, such that AB = BA = I, then B is called the inverse
matrix of A and it is denoted by A^{-1}. In that case A is said to be invertible. For example,

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