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Introduction

Matrices are one of the most powerful tools in mathematics.Matrices applications are used in business,finance,economics,genetics, sociology,modern,psychology and industrial management.

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Matrix:

A matrix is an ordered rectangular array of numbers(real or complex)or functions.The numbers or functions are called the elements or entries of the matrix. We denote matrices by capital letters.

It is to be noted that a matrix is just an arrangement of elements with out any value in rows and columns

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Order of a matrix:
A matrix having m rows and n columns is called a matrix of order mxn or simply mxn matrix(read as an m by n matrix).
Order of matrix mxn means there are m(rows) and n(columns). The number of elements in an mxn matrix will be equal to m n.
eg.

(i)

The order of this matrix is 2x2 (2rows and 2columns). Total number of element will be 4.(2x2)
(ii)

The order of this matrix is 3x3 (3rows and 3columns). Total number of elements are 9.(3x3)

Note: In general, if a matrix has order m x n then total number of elements in this matrix will be mn.

Type of Matrices

(i) Column Matrix : A matrix is said to be a column matrix or column vector if it has only one column. The order of column matrix is (m)x(1){m=rows and 1= column}
eg .

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(ii) Row Matrix : A matrix is said to be a row matrix or row vector if it has only one row. The order of row matrix is (1)x(m){1=row and m=columns}
eg .

(iii) Square Matrix :A matrix in which the number of rows are equal to number of column is said to be a square matrix. Thus an (m)x(n) matrix is said to be a square matrix if m = n and is known as a square matrix of order n.
eg .

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(iv) Diagonal Matrix :A square matrix is said to be a diagonal matrix if all its non diagonal elements are zero.
eg .

(v) Scalar Matrix : A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,
eg .

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(vi) Identity Matrix :A square matrix in which elements in the diagonal are all 1 and rest are all zero.
We denote the identity matrix of order n by I_n.
eg .

(vii) Zero Matrix or Null Matrix :A matrix is said to be zero matrix or null matrix if all elements are zero . We denote zero matrix by 0.
eg .

(viii) Rectangular Matrix :If the shape of matrix rectangular, then it is called as rectangular matrix .
eg .

(ix) Upper Triangular Matrix :A matrix is known as upper triangular matrix if all the elements below the leading diagonal or principal diagonal are zero.
eg .

(x) Lower Trangular Matrix :A matrix is known as lower triangular matrix if all the elements above the leading diagonal or principal diagonal are zero.
eg .

(xi) Sub Matrix :The matrix obtained by deleting one or more rows or columns or both of a matrix is called its sub matrix.
eg .
The sub matrix is obtained by deleting 3^rd row and I^st column.

(xii) Equal Matrices :Two matrices A = [a_ij] and B = [b_ij]are said to be equal it they full fill two conditions.
(a) The order of both the matrices is same.
(b) Corresponding elements in both the matrices are equal , that is a_ij = b_ij for all i and j symbolically, if two matrices A and B are equal , we write A = B
eg . (i)
(ii)

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Transpose of Matrix

If row and columns are interchanged then this process is called as transpose of matrix .It is denoted by A' or A^T or A̅ for example

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Properties of transpose of matrices:

(i)   (A')" = A

(ii)   (kA)' = kA' (where k is any constant)

(iii)   (A + B)' = A' + B'

(iv)   (AB)' = B'A'

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Multiplication of Matrix

Two matrix can be multiply only if the number of column of first matrix are equal to number of row of second matrix ,this condition is called as conformability of matrix.

Illustration

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Determinant

the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or. | A. |

Note : we can find the value of determinant by the method of expantion, one can expand determinwnt by six ways (along three rows and three column).

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Minor and Co-factors

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Singular Matrix: A matrix is singular if its determinant is zero, and its inverse will not exist

Non-Singular Matrix: A matrix is non-singular if its determinant is non - zero, and its inverse will exist

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Properties of Determinants

(1)   The value of the determinant remains unchanged if both rows and columns are interchanged.

(2)   If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

(3)   If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero.

(4)   If all elements of any row or coloumn will be zero then the value of determinant will also be zero.

(5)   If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

(6)   f some or all elements of a row or column of a determinant are expressed as the sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants. For example, Properties of Determinants.

(7)   If the equimultiples of corresponding elements of other rows (or columns) are added to every element of any row or column of a determinant, then the value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation R_i ? R_i + k R_j or C_i ? C_i+ k C_j .

(8)   If in determinant(order 3 or more) the elements in all the rows (or columns)are in A.P. with same or different common difference, the value of determinat is zero.

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