Summary Math 1st Year Chapter No.4

Chapter 4
DETERMINANTS




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™ All Mathematical truths are relative and conditional. — C.P. STEINMETZ ™
4.1 Introduction




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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




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matrices. This means, a system of linear equations like
a1 x + b1 y = c 1
pu a2 x + b2 y = c 2
⎡a b ⎤ ⎡ x⎤ ⎡c ⎤
can be represented as ⎢ 1 1 ⎥ ⎢ ⎥ = ⎢ 1 ⎥ . Now, this
be T

⎣ a2 b2 ⎦ ⎣ y ⎦ ⎣ c2 ⎦
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system of equations has a unique solution or not, is
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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
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a2 b2 (1749-1827)
equations has a unique solution). The number a1 b2 – a2 b1
⎡a b ⎤
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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
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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,
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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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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.




d
⎡a b ⎤ a b
If A = ⎢ ⎥ , then determinant of A is written as |A | = = det (A)
⎣c d ⎦ c d




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Remarks
(i) For matrix A, | A | is read as determinant of A and not modulus of A.
(ii) Only square matrices have determinants.




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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




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4.2.2 Determinant of a matrix of order two
⎡ a11 a12 ⎤
Let A= ⎢ ⎥ be a matrix of order 2 × 2,
⎣ a21 a22 ⎦
pu
then the determinant of A is defined as:
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det (A) = |A| = Δ = = a11a22 – a21a12
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2 4
Example 1 Evaluate .
–1 2
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2 4
Solution We have = 2 (2) – 4(–1) = 4 + 4 = 8.
–1 2
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x x +1
Example 2 Evaluate
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x –1 x
Solution We have
x x +1
= x (x) – (x + 1) (x – 1) = x2 – (x2 – 1) = x2 – x2 + 1 = 1
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x –1 x

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 105


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




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a11 a12 a13
i.e., | A | = a21 a22 a23




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a31 a32 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




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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,




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a22 a23
i.e., (–1)1 + 1 a11
a32 a33
Step 2 Multiply 2nd element a12 of R1 by (–1)1 + 2 [(–1)sum of suffixes in a12] and the second
pu
order determinant obtained by deleting elements of first row (R1) and 2nd column (C2)
of | A | as a12 lies in R1 and C2,
be T

a21 a23
i.e., (–1)1 + 2 a12
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a31 a33
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Step 3 Multiply third element a13 of R1 by (–1)1 + 3 [(–1)sum of suffixes in a ] and the second
13


order determinant obtained by deleting elements of first row (R1) and third column (C3)
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of | A | as a13 lies in R1 and C3,
a21 a22
C




i.e., (–1)1 + 3 a13 a a32
31

Step 4 Now the expansion of determinant of A, that is, | A | written as sum of all three
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terms obtained in steps 1, 2 and 3 above is given by
a22 a23 a a
det A = |A| = (–1)1 + 1 a11 a + (–1)1 + 2 a12 21 23
32 a33 a31 a33
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1+ 3 a21 a22
+ (–1) a13
a31 a32
or |A| = a11 (a22 a33 – a32 a23) – a12 (a21 a33 – a31 a23)
+ a13 (a21 a32 – a31 a22)

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= a11 a22 a33 – a11 a32 a23 – a12 a21 a33 + a12 a31 a23 + a13 a21 a32
– a13 a31 a22 ... (1)

$Note We shall apply all four steps together.




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Expansion along second row (R2)




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a 11 a 12 a 13
| A | = a 21 a 22 a 23
a 31 a 32 a 33




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Expanding along R2, we get

2 +1 a12 a13 a a
| A | = (–1) a21 + (–1)2 + 2 a22 11 13




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a32 a33 a31 a33

a11 a12
+ (–1) 2 + 3 a23
pu a31 a32
= – a21 (a12 a33 – a32 a13) + a22 (a11 a33 – a31 a13)
be T

– a23 (a11 a32 – a31 a12)
| A | = – a21 a12 a33 + a21 a32 a13 + a22 a11 a33 – a22 a31 a13 – a23 a11 a32
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o R


+ a23 a31 a12
= a11 a22 a33 – a11 a23 a32 – a12 a21 a33 + a12 a23 a31 + a13 a21 a32
tt E



– a13 a31 a22 ... (2)
Expansion along first Column (C1)
C




a11 a12 a13
| A | = a21 a22 a23
a31 a32 a33
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By expanding along C1, we get

1 + 1 a22 a23 a a13
| A | = a11 (–1) + a21 (−1) 2 + 1 12
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a32 a33 a32 a33

3 + 1 a 12 a13
+ a31 (–1) a22 a23
= a11 (a22 a33 – a23 a32) – a21 (a12 a33 – a13 a32) + a31 (a12 a23 – a13 a22)