# Education for Girl Child

Topics: Invertible matrix, Linear algebra, Determinant Pages: 58 (10481 words) Published: June 28, 2013
Chapter

4

DETERMINANTS
All Mathematical truths are relative and conditional. — C.P. STEINMETZ

4.1 Introduction
In the previous chapter, we have studied about matrices and algebra of matrices. We have also learnt that a system of algebraic equations can be expressed in the form of matrices. This means, a system of linear equations like a1 x + b1 y = c 1 a2 x + b2 y = c 2 ⎡ a b ⎤ ⎡ x ⎤ ⎡c ⎤ can be represented as ⎢ 1 1 ⎥ ⎢ ⎥ = ⎢ 1 ⎥ . Now, this ⎣ a2 b2 ⎦ ⎣ y ⎦ ⎣ c2 ⎦ system of equations has a unique solution or not, is determined by the number a1 b2 – a2 b1. (Recall that if

a1 b1 or, a1 b2 – a2 b1 ≠ 0, then the system of linear ≠ a2 b2 equations has a unique solution). The number a1 b2 – a2 b1

P.S. Laplace (1749-1827)

⎡a b ⎤ which determines uniqueness of solution is associated with the matrix A = ⎢ 1 1 ⎥ ⎣ a2 b2 ⎦ and is called the determinant of A or det A. Determinants have wide applications in Engineering, Science, Economics, Social Science, etc. In this chapter, we shall study determinants up to order three only with real entries. Also, we will study various properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle, adjoint and inverse of a square matrix, consistency and inconsistency of system of linear equations and solution of linear equations in two or three variables using inverse of a matrix.

4.2 Determinant
To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where aij = (i, j)th element of A.

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MATHEMATICS

This may be thought of as a function which associates each square matrix with a unique number (real or complex). If M is the set of square matrices, K is the set of numbers (real or complex) and f : M → K is defined by f (A) = k, where A ∈ M and k ∈ K, then f (A) is called the determinant of A. It is also denoted by | A | or det A or Δ. a b ⎡a b ⎤ If A = ⎢ ⎥ , then determinant of A is written as | A| = c d = det (A) ⎣c d ⎦ Remarks (i) For matrix A, | A | is read as determinant of A and not modulus of A. (ii) Only square matrices have determinants.

4.2.1 Determinant of a matrix of order one Let A = [a ] be the matrix of order 1, then determinant of A is defined to be equal to a 4.2.2 Determinant of a matrix of order two ⎡ a11 a12 ⎤ A= ⎢ ⎥ be a matrix of order 2 × 2, ⎣ a21 a22 ⎦ then the determinant of A is defined as:

Let

det (A) = |A| = Δ =
2 4 . –1 2

= a11a22 – a21a12

Example 1 Evaluate

Solution We have

2 4 = 2 (2) – 4(–1) = 4 + 4 = 8. –1 2 x x +1 x –1 x

Example 2 Evaluate Solution We have
x x –1 x +1 x

= x (x) – (x + 1) (x – 1) = x2 – (x2 – 1) = x2 – x2 + 1 = 1

4.2.3 Determinant of a matrix of order 3 × 3 Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order

DETERMINANTS

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3 corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and C3) giving the same value as shown below. Consider the determinant of square matrix A = [aij]3 × 3

a 11
i.e., | A | = a21 a31

a12 a22 a32

a13 a23 a33

Expansion along first Row (R1) Step 1 Multiply first element a11 of R1 by (–1)(1 + 1) [(–1)sum of suffixes in a11] and with the second order determinant obtained by deleting the elements of first row (R1) and first column (C1) of | A | as a11 lies in R1 and C1, i.e., (–1)1 + 1 a11

a22 a32

a23 a33

Step 2 Multiply 2nd element a12 of R1 by (–1)1 + 2 [(–1)sum of suffixes in a12] and the second order determinant obtained by deleting elements of first row (R1) and 2nd column (C2) of | A | as a12 lies in R1 and C2, i.e., (–1)1 + 2 a12

a21 a23 a31 a33
13

Step 3 Multiply third element a13 of R1 by (–1)1 + 3 [(–1)sum of suffixes in a ] and the second order determinant obtained by deleting elements of...