NCERT Textbook: Determinants - Notes | Study Mathematics (Maths) Class 12 - JEE

 Page 1


? 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
a
1
 x + b
1
 y = c
1
a
2
 x + b
2
 y = c
2
can be represented as 
11 1
22 2
ab c x
ab c y
? ? ?? ??
=
? ? ?? ??
?? ? ? ??
. Now, this
system of equations has a unique solution or not, is
determined by the number a
1
 b
2
 – a
2 
b
1
. (Recall that if
11
22
ab
ab
? or, a
1
 b
2
 – a
2 
b
1
 ? 0,  then the system of linear
equations has a unique solution). The number a
1
 b
2
 – a
2 
b
1
which determines uniqueness of solution is associated with the matrix 
11
22
A
ab
ab
??
=
??
??
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 = [ a
ij
] of order n, we can associate a number (real or
complex) called determinant of the square matrix A, where a
ij
 = (i, j)
th
 element of A.
Chapter 4
DETERMINANTS
P.S. Laplace
(1749-1827)
2022-23
Page 2


? 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
a
1
 x + b
1
 y = c
1
a
2
 x + b
2
 y = c
2
can be represented as 
11 1
22 2
ab c x
ab c y
? ? ?? ??
=
? ? ?? ??
?? ? ? ??
. Now, this
system of equations has a unique solution or not, is
determined by the number a
1
 b
2
 – a
2 
b
1
. (Recall that if
11
22
ab
ab
? or, a
1
 b
2
 – a
2 
b
1
 ? 0,  then the system of linear
equations has a unique solution). The number a
1
 b
2
 – a
2 
b
1
which determines uniqueness of solution is associated with the matrix 
11
22
A
ab
ab
??
=
??
??
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 = [ a
ij
] of order n, we can associate a number (real or
complex) called determinant of the square matrix A, where a
ij
 = (i, j)
th
 element of A.
Chapter 4
DETERMINANTS
P.S. Laplace
(1749-1827)
2022-23
 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 ?.
If A = 
ab
cd
??
??
??
, then determinant of A is written as |A| = 
ab
cd
 = det (A)
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
Let A = 
11 12
21 22
aa
aa
??
??
??
 be a matrix of order 2 × 2,
then the determinant of A is defined as:
det (A) = |A| = ? =  = a
11
a
22
 – a
21
a
12
Example 1 Evaluate 
24
–1 2
.
Solution We have 
24
–1 2
 = 2 (2) – 4(–1) = 4 + 4 = 8.
Example 2 Evaluate 
1
–1
xx
xx
+
Solution We have
1
–1
xx
xx
+
 = x (x) – (x + 1) (x – 1)  = x
2
 – (x
2
 – 1) = x
2
 – x
2
 + 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
2022-23
Page 3


? 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
a
1
 x + b
1
 y = c
1
a
2
 x + b
2
 y = c
2
can be represented as 
11 1
22 2
ab c x
ab c y
? ? ?? ??
=
? ? ?? ??
?? ? ? ??
. Now, this
system of equations has a unique solution or not, is
determined by the number a
1
 b
2
 – a
2 
b
1
. (Recall that if
11
22
ab
ab
? or, a
1
 b
2
 – a
2 
b
1
 ? 0,  then the system of linear
equations has a unique solution). The number a
1
 b
2
 – a
2 
b
1
which determines uniqueness of solution is associated with the matrix 
11
22
A
ab
ab
??
=
??
??
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 = [ a
ij
] of order n, we can associate a number (real or
complex) called determinant of the square matrix A, where a
ij
 = (i, j)
th
 element of A.
Chapter 4
DETERMINANTS
P.S. Laplace
(1749-1827)
2022-23
 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 ?.
If A = 
ab
cd
??
??
??
, then determinant of A is written as |A| = 
ab
cd
 = det (A)
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
Let A = 
11 12
21 22
aa
aa
??
??
??
 be a matrix of order 2 × 2,
then the determinant of A is defined as:
det (A) = |A| = ? =  = a
11
a
22
 – a
21
a
12
Example 1 Evaluate 
24
–1 2
.
Solution We have 
24
–1 2
 = 2 (2) – 4(–1) = 4 + 4 = 8.
Example 2 Evaluate 
1
–1
xx
xx
+
Solution We have
1
–1
xx
xx
+
 = x (x) – (x + 1) (x – 1)  = x
2
 – (x
2
 – 1) = x
2
 – x
2
 + 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
2022-23
DETERMINANTS     105
3 corresponding to each of three rows (R
1
, R
2
 and R
3
) and three columns (C
1
, C
2
 and
C
3
) giving the same value as shown below.
Consider the determinant of square matrix A = [a
ij
]
3 × 3
i.e., | A | =
21 22 23
31 32 33
aa a
aa a
11 12 13
aa a
Expansion along first Row (R
1
)
Step 1 Multiply first element a
11
 of R
1
 by (–1)
(1 + 1)
 [(–1)
sum of suffixes in a
11] and with the
second order determinant obtained by deleting the elements of first row (R
1
) and first
column (C
1
) of | A | as a
11
 lies in R
1
 and C
1
,
i.e., (–1)
1 + 1
 a
11
 
22 23
32 33
aa
aa
Step 2 Multiply 2nd element a
12
 of R
1
 by (–1)
1 + 2
 [(–1)
sum of suffixes in a
12] and the second
order determinant obtained by deleting elements of first row (R
1
) and 2nd column (C
2
)
of | A | as a
12
 lies in R
1
 and C
2
,
i.e., (–1)
1 + 2
a
12
 
21 23
31 33
aa
aa
Step 3 Multiply third element a
13
 of R
1
 by (–1)
1 + 3
 [(–1)
sum of suffixes in a
13
] and the second
order determinant obtained by deleting elements of first row (R
1
) and third column (C
3
)
of | A | as a
13
 lies in R
1
 and C
3
,
i.e., (–1)
1 + 3
a
13
 
21 22
31 32
aa
aa
Step 4 Now the expansion of determinant of A, that is, | A | written as sum of all three
terms obtained in steps 1, 2 and 3 above is given by
det A = |A| = (–1)
1 + 1
 a
11
 
22 23 21 23 12
12
32 33 31 33
(–1)
aa aa
a
aa aa
+
+
         + 
21 22 13
13
31 32
(–1)
aa
a
aa
+
or |A| = a
11
 ( a
22
 a
33
 – a
32 
a
23
) – a
12
 ( a
21
 a
33
 – a
31 
a
23
)
+ a
13
 ( a
21
 a
32
 – a
31
 a
22
)
2022-23
Page 4


? 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
a
1
 x + b
1
 y = c
1
a
2
 x + b
2
 y = c
2
can be represented as 
11 1
22 2
ab c x
ab c y
? ? ?? ??
=
? ? ?? ??
?? ? ? ??
. Now, this
system of equations has a unique solution or not, is
determined by the number a
1
 b
2
 – a
2 
b
1
. (Recall that if
11
22
ab
ab
? or, a
1
 b
2
 – a
2 
b
1
 ? 0,  then the system of linear
equations has a unique solution). The number a
1
 b
2
 – a
2 
b
1
which determines uniqueness of solution is associated with the matrix 
11
22
A
ab
ab
??
=
??
??
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 = [ a
ij
] of order n, we can associate a number (real or
complex) called determinant of the square matrix A, where a
ij
 = (i, j)
th
 element of A.
Chapter 4
DETERMINANTS
P.S. Laplace
(1749-1827)
2022-23
 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 ?.
If A = 
ab
cd
??
??
??
, then determinant of A is written as |A| = 
ab
cd
 = det (A)
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
Let A = 
11 12
21 22
aa
aa
??
??
??
 be a matrix of order 2 × 2,
then the determinant of A is defined as:
det (A) = |A| = ? =  = a
11
a
22
 – a
21
a
12
Example 1 Evaluate 
24
–1 2
.
Solution We have 
24
–1 2
 = 2 (2) – 4(–1) = 4 + 4 = 8.
Example 2 Evaluate 
1
–1
xx
xx
+
Solution We have
1
–1
xx
xx
+
 = x (x) – (x + 1) (x – 1)  = x
2
 – (x
2
 – 1) = x
2
 – x
2
 + 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
2022-23
DETERMINANTS     105
3 corresponding to each of three rows (R
1
, R
2
 and R
3
) and three columns (C
1
, C
2
 and
C
3
) giving the same value as shown below.
Consider the determinant of square matrix A = [a
ij
]
3 × 3
i.e., | A | =
21 22 23
31 32 33
aa a
aa a
11 12 13
aa a
Expansion along first Row (R
1
)
Step 1 Multiply first element a
11
 of R
1
 by (–1)
(1 + 1)
 [(–1)
sum of suffixes in a
11] and with the
second order determinant obtained by deleting the elements of first row (R
1
) and first
column (C
1
) of | A | as a
11
 lies in R
1
 and C
1
,
i.e., (–1)
1 + 1
 a
11
 
22 23
32 33
aa
aa
Step 2 Multiply 2nd element a
12
 of R
1
 by (–1)
1 + 2
 [(–1)
sum of suffixes in a
12] and the second
order determinant obtained by deleting elements of first row (R
1
) and 2nd column (C
2
)
of | A | as a
12
 lies in R
1
 and C
2
,
i.e., (–1)
1 + 2
a
12
 
21 23
31 33
aa
aa
Step 3 Multiply third element a
13
 of R
1
 by (–1)
1 + 3
 [(–1)
sum of suffixes in a
13
] and the second
order determinant obtained by deleting elements of first row (R
1
) and third column (C
3
)
of | A | as a
13
 lies in R
1
 and C
3
,
i.e., (–1)
1 + 3
a
13
 
21 22
31 32
aa
aa
Step 4 Now the expansion of determinant of A, that is, | A | written as sum of all three
terms obtained in steps 1, 2 and 3 above is given by
det A = |A| = (–1)
1 + 1
 a
11
 
22 23 21 23 12
12
32 33 31 33
(–1)
aa aa
a
aa aa
+
+
         + 
21 22 13
13
31 32
(–1)
aa
a
aa
+
or |A| = a
11
 ( a
22
 a
33
 – a
32 
a
23
) – a
12
 ( a
21
 a
33
 – a
31 
a
23
)
+ a
13
 ( a
21
 a
32
 – a
31
 a
22
)
2022-23
 106 MATHEMATICS
= a
11
a
22
 a
33
 – a
11
 a
32 
a
23
 – a
12
 a
21
a
33
 + a
12
 a
31 
a
23
 + a
13
 a
21
 a
32
– a
13
 a
31
 a
22
... (1)
?
Note  We shall apply all four steps together.
Expansion along second row (R
2
)
| A | =
11 12 13
31 32 33
aa a
aa a
21 22 23
aa a
Expanding along R
2
,we get
| A | =
12 13 11 13 21 22
21 22
32 33 31 33
(–1) (–1)
aa aa
a a
aa aa
+ +
+
11 12 23
23
31 32
(–1)
aa
a
aa
+
+
= – a
21
 ( a
12
 a
33
 – a
32 
a
13
) + a
22
 ( a
11
 a
33
 – a
31 
a
13
)
– a
23
 ( a
11
 a
32
 – a
31
 a
12
)
| A | = – a
21
 a
12
 a
33
 + a
21
 a
32 
a
13
 + a
22
 a
11
 a
33
 – a
22
 a
31 
a
13
 – a
23
 a
11
 a
32
 + a
23
 a
31
 a
12
= a
11
 a
22
 a
33
 – a
11
 a
23
 a
32
 – a
12
 a
21
 a
33
 + a
12
 a
23
 a
31
 + a
13
 a
21
 a
32
– a
13
 a
31
 a
22
... (2)
Expansion along first Column (C
1
)
| A | =
12 13
22 23
32 33
11
21
31
a
a
a
aa
aa
aa
By expanding along C
1
, we get
| A | =
22 23 12 13 11 21
11 21
32 33 32 33
(–1) ( 1)
aa aa
a a
aa aa
+ +
+-
+ 
12 13 31
31
22 23
(–1)
aa
a
aa
+
= a
11
 (a
22
a
33
 – a
23
 a
32
) – a
21
 (a
12
a
33
 – a
13
 a
32
) + a
31
 (a
12
 a
23
 – a
13
 a
22
)
2022-23
Page 5


? 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
a
1
 x + b
1
 y = c
1
a
2
 x + b
2
 y = c
2
can be represented as 
11 1
22 2
ab c x
ab c y
? ? ?? ??
=
? ? ?? ??
?? ? ? ??
. Now, this
system of equations has a unique solution or not, is
determined by the number a
1
 b
2
 – a
2 
b
1
. (Recall that if
11
22
ab
ab
? or, a
1
 b
2
 – a
2 
b
1
 ? 0,  then the system of linear
equations has a unique solution). The number a
1
 b
2
 – a
2 
b
1
which determines uniqueness of solution is associated with the matrix 
11
22
A
ab
ab
??
=
??
??
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 = [ a
ij
] of order n, we can associate a number (real or
complex) called determinant of the square matrix A, where a
ij
 = (i, j)
th
 element of A.
Chapter 4
DETERMINANTS
P.S. Laplace
(1749-1827)
2022-23
 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 ?.
If A = 
ab
cd
??
??
??
, then determinant of A is written as |A| = 
ab
cd
 = det (A)
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
Let A = 
11 12
21 22
aa
aa
??
??
??
 be a matrix of order 2 × 2,
then the determinant of A is defined as:
det (A) = |A| = ? =  = a
11
a
22
 – a
21
a
12
Example 1 Evaluate 
24
–1 2
.
Solution We have 
24
–1 2
 = 2 (2) – 4(–1) = 4 + 4 = 8.
Example 2 Evaluate 
1
–1
xx
xx
+
Solution We have
1
–1
xx
xx
+
 = x (x) – (x + 1) (x – 1)  = x
2
 – (x
2
 – 1) = x
2
 – x
2
 + 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
2022-23
DETERMINANTS     105
3 corresponding to each of three rows (R
1
, R
2
 and R
3
) and three columns (C
1
, C
2
 and
C
3
) giving the same value as shown below.
Consider the determinant of square matrix A = [a
ij
]
3 × 3
i.e., | A | =
21 22 23
31 32 33
aa a
aa a
11 12 13
aa a
Expansion along first Row (R
1
)
Step 1 Multiply first element a
11
 of R
1
 by (–1)
(1 + 1)
 [(–1)
sum of suffixes in a
11] and with the
second order determinant obtained by deleting the elements of first row (R
1
) and first
column (C
1
) of | A | as a
11
 lies in R
1
 and C
1
,
i.e., (–1)
1 + 1
 a
11
 
22 23
32 33
aa
aa
Step 2 Multiply 2nd element a
12
 of R
1
 by (–1)
1 + 2
 [(–1)
sum of suffixes in a
12] and the second
order determinant obtained by deleting elements of first row (R
1
) and 2nd column (C
2
)
of | A | as a
12
 lies in R
1
 and C
2
,
i.e., (–1)
1 + 2
a
12
 
21 23
31 33
aa
aa
Step 3 Multiply third element a
13
 of R
1
 by (–1)
1 + 3
 [(–1)
sum of suffixes in a
13
] and the second
order determinant obtained by deleting elements of first row (R
1
) and third column (C
3
)
of | A | as a
13
 lies in R
1
 and C
3
,
i.e., (–1)
1 + 3
a
13
 
21 22
31 32
aa
aa
Step 4 Now the expansion of determinant of A, that is, | A | written as sum of all three
terms obtained in steps 1, 2 and 3 above is given by
det A = |A| = (–1)
1 + 1
 a
11
 
22 23 21 23 12
12
32 33 31 33
(–1)
aa aa
a
aa aa
+
+
         + 
21 22 13
13
31 32
(–1)
aa
a
aa
+
or |A| = a
11
 ( a
22
 a
33
 – a
32 
a
23
) – a
12
 ( a
21
 a
33
 – a
31 
a
23
)
+ a
13
 ( a
21
 a
32
 – a
31
 a
22
)
2022-23
 106 MATHEMATICS
= a
11
a
22
 a
33
 – a
11
 a
32 
a
23
 – a
12
 a
21
a
33
 + a
12
 a
31 
a
23
 + a
13
 a
21
 a
32
– a
13
 a
31
 a
22
... (1)
?
Note  We shall apply all four steps together.
Expansion along second row (R
2
)
| A | =
11 12 13
31 32 33
aa a
aa a
21 22 23
aa a
Expanding along R
2
,we get
| A | =
12 13 11 13 21 22
21 22
32 33 31 33
(–1) (–1)
aa aa
a a
aa aa
+ +
+
11 12 23
23
31 32
(–1)
aa
a
aa
+
+
= – a
21
 ( a
12
 a
33
 – a
32 
a
13
) + a
22
 ( a
11
 a
33
 – a
31 
a
13
)
– a
23
 ( a
11
 a
32
 – a
31
 a
12
)
| A | = – a
21
 a
12
 a
33
 + a
21
 a
32 
a
13
 + a
22
 a
11
 a
33
 – a
22
 a
31 
a
13
 – a
23
 a
11
 a
32
 + a
23
 a
31
 a
12
= a
11
 a
22
 a
33
 – a
11
 a
23
 a
32
 – a
12
 a
21
 a
33
 + a
12
 a
23
 a
31
 + a
13
 a
21
 a
32
– a
13
 a
31
 a
22
... (2)
Expansion along first Column (C
1
)
| A | =
12 13
22 23
32 33
11
21
31
a
a
a
aa
aa
aa
By expanding along C
1
, we get
| A | =
22 23 12 13 11 21
11 21
32 33 32 33
(–1) ( 1)
aa aa
a a
aa aa
+ +
+-
+ 
12 13 31
31
22 23
(–1)
aa
a
aa
+
= a
11
 (a
22
a
33
 – a
23
 a
32
) – a
21
 (a
12
a
33
 – a
13
 a
32
) + a
31
 (a
12
 a
23
 – a
13
 a
22
)
2022-23
DETERMINANTS     107
| A | = a
11
a
22
 a
33
 – a
11
 a
23
 a
32
 – a
21
 a
12
 a
33
 + a
21
 a
13
 a
32
 + a
31
 a
12
 a
23
– a
31
 a
13
 a
22
= a
11
 a
22
 a
33
 – a
11
 a
23
 a
32
 – a
12
 a
21
 a
33
 + a
12
 a
23
 a
31
 + a
13
 a
21
 a
32
– a
13
 a
31
 a
22
... (3)
Clearly, values of |A | in (1), (2) and (3) are equal. It is left as an exercise to the
reader to verify that the values of |A| by expanding along R
3
, C
2
 and C
3
 are equal to the
value of |A| obtained in (1), (2) or (3).
Hence, expanding a determinant along any row or column gives same value.
Remarks
(i) For easier calculations, we shall expand the determinant along that row or column
which contains maximum number of zeros.
(ii) While expanding, instead of multiplying by (–1)
i + j
, we can multiply by +1 or –1
according as (i + j) is even or odd.
(iii) Let A = 
22
40
??
??
??
 and B = 
11
20
??
??
??
 . Then, it is easy to verify that A = 2B. Also
|A | = 0 – 8 = – 8 and |B | = 0 – 2 = – 2.
Observe that, |A | = 4(– 2) = 2
2
|B| or |A| = 2
n
|B |, where n = 2 is the order of
square matrices A and B.
In general, if A = kB where A and B are square matrices of order n, then | A| = k
n
| B |, where n = 1, 2, 3
Example 3 Evaluate the determinant  ? = 
124
–1 3 0
410
.
Solution Note that in the third column, two entries are zero. So expanding along third
column (C
3
), we get
? =
–13 12 1 2
4 –0 0
4 1 4 1 –1 3
+
= 4 (–1 – 12) – 0 + 0  = – 52
Example 4 Evaluate ? = 
0 sin – cos
– sin 0 sin
cos – sin 0
aa
a ß
aß
.
2022-23