Education for Girl Child
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.
104
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
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.
104
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