NCERT Book Class 12 Maths Chapter 4 Determinants

76 MATHEMATICS

Chapter 4
DETERMINANTS

v All Mathematical truths are relative and conditional. — C.P. STEINMETZ v
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 P.S. Laplace
≠ or, a1 b2 – a2 b1 ≠ 0, then the system of linear
a2 b2 (1749-1827)
equations has a unique solution). The number a1 b2 – a2 b1
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.