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complex) called determinant of the square matrix A, where aij = (i, j)th element of Step 2 Multiply 2nd element a12 of R1 by (–1)1 + 2 [(–1)sum of suffixes in a In earlier classes, we have studied that the area of a triangle whose vertices are

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The knowledge of matrices is necessary in various branches of mathematics Solution In general a 3 × 2 matrix is given by 11 12 21 22 31 32 A a a a a

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It is denoted by det A or A for a square matrix A The determinant of the (2 x 2) matrix A = 11 12 21 22

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Every square matrix A is associated with a number, called its determinant and it is denoted by det (A) or A Only square matrices have determinants The matrices  

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16 Singular Matrix A square matrix A is said to be singular matrix, if determinant of A denoted by det (A) or A is 

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m × n matrix, the element a ij belongs to the ith row and jth column 11 12 13 In this lesson, we will learn various properties of determinants and also In earlier classes, you have learnt how to solve linear equations in two or three unknowns

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Chapter 9 222 Matrices and Determinants

Chapter 9 Matrices and Determinants

9.1 Introduction:

In many economic analysis, variables are assumed to be related by sets of linear equations. Matrix algebra provides a clear and concise notation for the formulation and solution of such problems, many of which would be complicated in conventional algebraic notation. The concept of determinant and is based on that of matrix. Hence we shall first explain a matrix.

9.2 Matrix:

A set of mn numbers (real or complex), arranged in a rectangular formation (array or table) having m rows and n columns and enclosed by a square bracket [ ] is called m . An m n matrix is expressed as

11 12 1n

21 22 2n

m1 m2 mn A= a a a a a a a a a

The letters a

ij stand for real numbers. Note that aij is the element in the ith row and jth column of the matrix .Thus the matrix A is sometimes denoted by simplified form as (a ij) or by {aij} i.e., A = (aij) Matrices are usually denoted by capital letters A, B, C etc and its elements by small letters a, b, c etc.

Order of a Matrix:

The order or dimension of a matrix is the ordered pair having as first component the number of rows and as second component the number of columns in the matrix. If there are 3 rows and 2 columns in a matrix, then its order is written as (3, 2) or (3 x 2) read as three by two. In general if m are rows and n are columns of a matrix, then its order is (m x n).

Examples:

Chapter 9 223 Matrices and Determinants

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

a a a a1b b b b1 2 3 , 2 and c c c c4 5 63d d d d are matrices of orders (2 x 3), (3 x 1) and (4 x 4) respectively.

9.3 Some types of matrices:

1. Row Matrix and Column Matrix:

A matrix consisting of a single row is called a row matrix or a row vector, whereas a matrix having single column is called a column matrix or a column vector.

2. Null or Zero Matrix:

A matrix in which each element is called a Null or Zero matrix. Zero matrices are generally denoted by the symbol O. This distinguishes zero matrix from the real number 0.

For example O =

0000 0000

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