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