Chapter 2: Determinants 1
SECTION A Determinant of a Matrix
By the end of this section you will be able to
 evaluate the determinant of various square matrices
 understand what is meant by the terms cofactor and minor of a matrix
 determine the inverse of a square matrix
T. Seki was the first person to study determinants which arose naturally out of a system of linear equations and he highlighted this work in 1683.
Seki taught himself mathematics from a very young age but was initially introduced to the subject by a household servant. He was from a family of Samurai warriors.
Lewis Carroll (or to use his real name Charles Dodgson) wrote a book on determinants called ‘An Elementary Theory of
Determinants’ in 1867. Clearly he is better known for his popular book ‘Alice’s Adventures in Wonderland’.
Fig 1 T. Seki 1642 to 1708
Lewis Carroll proposed that Oxford University set up a Mathematical
Institute 65 years before it was eventually built. He also wrote to the Dean proposing his salary be lowered from £300 to £200 per year because his Oxford College was suffering a financial crisis.
In this chapter we associate a number for every square matrix. Can you remember what is meant by a square matrix?
A square matrix is a matrix where the number of rows and columns are equal , that is a matrix of size n by n or n n . The number associated with each square matrix is called the determinant of the matrix and tells us whether the matrix is invertible or not. Generally in this chapter, a matrix will mean a square matrix.
A1 Determinant and Inverse of a 2 by 2 Matrix
We first find the determinant of a 2 2 matrix and then expand to 3 3, , n n size matrices.
Example 1
Consider the general 2 2 matrix A
What do you notice about your result?
 a b c d


and the matrix B d
 b
  c a


. Evaluate AB .
Solution
AB
 a b
 d
 b c d   c a


 

 ad
 bc
  cd
 cd
  ad




 ad
 bc
0
0 ad
 bc



Taking out a common factor ad bc
 ad
 bc



1 0
0 1


  ad
 bc

I
The matrix multiplication AB gives a multiple ad
 bc of the identity matrix, I . This

Chapter 2: Determinants multiple
 ad
 bc

is called the determinant of matrix
The determinant of a matrix A is normally denoted by
A
 a b c d


. det
 
or A and is a scalar not a matrix.
Hence the determinant of the general 2 2
   ad
 bc
matrix A
  b

 a b


is defined as
(2.1) det A a minus d c c d
What does this formula (2.1) mean?
It means the determinant of a 2 2 matrix is the result of multiplying the entries of the leading diagonal and subtracting the product of the other diagonal. Remember the leading diagonal are the entries of the matrix which slope downwards to the right, .
Example 2
Again consider the 2 2 matrices A
 a b c d


and B
Evaluate the matrix multiplication AB provided det
 det
1
 

 d
 b
 c a


.

0 .
What do you notice about your result?
Solution
AB


 a b c d


 det
1
 


 d
 b c a



 det
1
 

 a b
 d
 b c d 
 c a

 det
1
   ad
 bc

I

Remember

 where k

A
 
1/ det
 
 k
 
is a scalar




By Example 1 because we have the same matrices


 det
1
   

By (2.1) we have

 det
   ad
 bc

0




I
Note that the matrix multiplication AB
 Cancelling Out det gives the identity matrix I

1 0
0 1


.
Since AB

I , what conclusions can we draw about the matrices A and B?
The given matrix A
 

 a b c d


has an inverse matrix A

1   det
1
 

 d
 b
 c a

 we have AB

I which means B is the inverse of matrix A , that is
Hence the inverse of the general 2 2 matrix A
 a b c d


B

is given by
A

1
.
(2.2) A

1  det
1
 

 d
 b
 c a


provided det

0 because
What does this formula mean?
2

Chapter 2: Determinants
The inverse of a 2 2 matrix is determined by interchanging entries along the leading diagonal and placing a negative sign in the other and then multiplying this matrix by
1 det
 
.
What can we say if the determinant is zero, that is
If det det
  
0 ?
  
0 then the matrix A is non-invertible (singular), it has no inverse .
Example 3
Find the inverses of the following matrices:
(a) A

2 3

1 5


(b) B

 
2 1

1 2


(c) C
 

 


Solution
(a) Before we can find the inverse we need to evaluate the determinant. Why?
Because if the determinant is 0 then the matrix does not have an inverse. Therefore by
(2.1) det

 a b c d


 ad
 bc minus we have det

The inverse matrix det


2 3

1 5


  
1 3

13
A

1
is given by the above formula (2.2) with det

13 :
A

1 

 
2 3
1 5



1

1
13


5

3
1 2




By (2.2)

 a b c d



1
 det
1
 

 d
 b
 c a




(b) We adopt the same procedure as part (a) to find B

1
. By
(2.1) det

 a b c d


 ad
 bc minus we have det
By substituting det
 det


2 1

 

2

2
 
1 1

2 1 3

1 2
  
3 into the inverse formula (2.2) we have
B

1 


2 1

1 2



1

1
3


1
2

1
2




By (2.2)

 a b c d



1
 det
1
 

 d
 b
 c a





(c) Similarly applying (2.1) det
 ad
 bc we have det
 det


 
 


         
0
What can we conclude about the matrix C?
Since det
  
0 therefore the matrix C is non-invertible (singular). This means it does not have an inverse.
A2 Applications to Transformations
Example 4
3

Chapter 2: Determinants
Consider a triangle given by the coordinates P
 
Q

2, 0

and R

0, 3

. Let the matrix A represent this triangle PQR and determine the image of this triangle under the transformation given by BA where B

2 0
3 4


.
By illustrating this transformation determine the areas of the triangle PQR and the transformed triangle P’Q’R’ . How does this transformation change the size of the area?
Solution
We are given coordinates P
 
Q

2, 0

and R

0, 3

therefore A

0 2 0
0 0 3


.
Evaluating the matrix multiplication BA :
P Q R P ' Q ' R '
BA



2 0

0 2 0
 
0 4 0
3 4 0 0 3 0 6 12

 y
Plotting this:
14
12
10
8
R'
R
6 Q'
Fig 1 P and P'
4
2
1 2
Q
3 4 5 x
The area of shaded triangle PQR
 
3 and area of large triangle ' ' '
2
The transformation B increases the area by a factor of 8 because 24 / 3 8 .

This factor 8 is called the determinant of the matrix B and it is evaluated by
Determinant of matrix


2 0
3 4



2 4
 
3 0

2

24 .
Before we discuss the inverse of a 3 by 3 matrix, we examine what is meant by the terms
‘Minors’ and ‘Cofactors’.
A2 Minors and Cofactors
Consider the general 3 by 3 matrix A
 


 a d e f g b h c i


. The determinant of the remaining matrix after deleting the row and column of an entry is called the minor of that entry.
For example, in the case of the matrix A we have det

 e f h i


is the minor of entry a
4

Chapter 2: Determinants
What is the minor of entry c? det

 d f g i


is the minor of entry b det

 d e g h


What is the minor of entry e?

 a e c


Deleting the row and column containing the entry e. g i
Hence det

 a c g i


is the minor of entry e .
Example 4
Determine the minor of

1 in



3 5 7
1 2 3


 
4 4

9 
Solution
After deleting the rows and columns containing

1 ,



1
5 7


 4

9 
Deleting these. we obtain the matrix


5 7
4

9 det




. The minor of
5 7
4

9



By (2.1)

5

1 is the determinant of this matrix:
   
4 7
  
73
Next we give the general definition of minor .
Definition (2.3). Consider a square matrix A . Let a ij
be the entry in the ith row and jth column of matrix A . The minor M of entry ij a ij
is the determinant of the remaining matrix after deleting the entries in the ith row and jth column.
M ij
 det

 a
11 a ij a
1 n

 ith row
 jth column
 a n 1 a nn

matrix we will need to find the determinant of a 2 2 matrix as seen in the above Example 4. For entries in a
3 3 matrix which we have not
4 4 yet stated.
matrix we will need to find the determinant of a
Next we define the term cofactor which is a number associated with the minor M ij
.
Definition (2.4). Consider a square matrix A . Let a ij
be the entry in the ith row and jth column of matrix A . The cofactor C ij
of the entry a ij
is defined as
C ij
M ij
5

Chapter 2: Determinants where M is the minor of entry ij a ij
.
The cofactor is given by C ij
  det

 a
11 a ij a
1 n


. a n 1 a nn
This definition might be difficult to follow because of the ij notation but this ij only locates the entry of a matrix. There is no easier way to locate an entry.
The cofactor is just the minor of an entry with a plus or minus sign depending on the entry.
For example the cofactor of the first entry a
11
is equal to the minor, C
11

M
11
because
   
2 
1 . What is the cofactor of the entry a ?
12
In this case a
12 means i

1 , j

2 and i j 1 2 3 therefore C
12
 
3
M
12
 
M
12
.
What is the cofactor of the entry
Since i j 1 3 4 a ?
13
therefore C
13
 
4
M
13

M
13
. If we carry on developing the cofactors we find they are just the minors with a place sign. Generally for a n n matrix we have minors with the following place signs:






  
  
  






What do you notice about the place signs?
The first entry of a matrix has a positive sign and then the place signs alternate .
Example 5
Determine the cofactor of 5 in


3 5 7

1 2 3




4 4

9 
Solution
After deleting the row and column containing 5 we have the minor of 5 is det




1
4

3
9


By (2.1)

1
   
4 3

21


(2.1) det

 a c b d


 ad
 bc


According to the rule, the place sign is negative, so the cofactor of 5 is

21
Note that the minor of 5 is 21 but the cofactor is

21
.
because the position of 5 in the matrix gives it a negative sign.
We can write the determinant of a 3 3 matrix in terms of its cofactors, that is if A is a general 3 3 matrix:
A
 


 a g b h i c d e f






Remember place signs are


  
   
 
 
 then
(2.5) det cofactor of a
  cofactor of b
  cofactor of c

Expanding out (2.5) gives
6

Chapter 2: Determinants 7
(2.6) det
 a 

 det

 e h i f


 
  b 
  det

 d f g i
 
 
Why is there a minus sign in front of the b in formula (2.6)? c


  det

 d e g h





This is no mistake , the minus sign comes about because the place sign for b is minus.
We can find the determinant of a matrix by expanding along any of the rows or any of the columns. For example, the formula for expanding along the second row is det
   d
 cofactor of d
  cofactor of e
  f
 cofactor of f

What is the formula for expanding along the bottom row? det
   g
 cofactor of g
  cofactor of h
  cofactor of i

We can also expand along any column. The formula for expanding along the first column is det
   cofactor of a
  cofactor of d
  cofactor of g

The general formula for the determinant of a n n matrix A






 a a
11 12 a a
21 22 a n 1 a n 2 n

3 is defined as a
1 n a
2 n a nn





 where
(2.7) det
 a C
11 11
 a C
12 12
 a C
13 13
  a C
1 n 1 n
 k n 

1 a C
1 k 1 k where the a
’s are the entries of the given matrix and
C
’s are the corresponding cofactors.
Don’t be put off by the sigma 
notation. This is just a compact way of writing the above sum given to us by the great Swiss mathematician Euler, pronounced ‘Oiler’, (1707 to 1783). n 
means summing k

1 a C
1 k 1 k a C from k

1 to k
 n .
1 k 1 k
(2.7) is the formula of the determinant for expanding along the first row of the matrix A .
What is the formula for expanding along the ith row?
For expanding along the ith row of the matrix, the formula is
(2.8) det
 a C i 1 i 1
 a C i 2 i 2
 a C i 3 i 3
  a C in in
 k n 

1 a C ik ik
This is sometimes called the Laplacian Expansion named after the French mathematician Pierre
Laplace (1749 to 1827).
Of course you can write a formula of the determinant for expanding along the jth column.
Example 6
Find the determinant of
A




1

6 3
5 6

7


 
2 0 1 
Solution
Which row or column should we expand along?
Since there is a 0 in the bottom row it is easier to expand along this row:

Chapter 2: Determinants det



1
5

6
6
3
7


2 det
 
2 0 1 



6 3
6

7



0 det
Expanding along this row
        
24
By (2.1) By (2.1)



1 3
5

7



1det



1

6
5 6


Note that the middle term on the Right-Hand Side is zero so we just need to evaluate the other two terms. We could have also found the determinant of A by expanding along the second column because this also contains (the same) zero.
In general if a row or column contains zero(s) then expanding along that row or column makes the arithmetic easier.
We can also obtain the determinant of a 4 4, 5 5, 6 6 etc matrix but it becomes very laborious to do this just using pen and paper unless we can establish zeros in the matrix. In these cases it is more convenient to use a graphical calculator or MATLAB.
The MATLAB command for finding the determinant of a matrix A is det(A).
A3 Cofactor Matrix
Let C be the new matrix consisting of the cofactors of the general matrix A . If
A

 a b c d e f


then C


A
D
B
E
C
F

 g h i G H I where A is the cofactor of a , B is the cofactor of b , C is the cofactor of c etc. The matrix C is called the cofactor matrix and it is used in finding the inverse matrix. Note that bold C is the cofactor matrix and plain C is the cofactor of the entry c .
Example 7
Find the cofactor matrix C of
A


1

1 5
3 9 7




2 1 0 




Remember the place signs are



  
  
 
 

Solution
Cofactor of the first entry, 1, is det


9 7
1 0


Cofactor of

1 is

Minus place sign det


3 7

2 0

By (2.1)


By (2.1)

 
9 0

 
3 0
1 7
 
  
2 7

7
 
14
Cofactor of 5 is det


3 9

2 1



By (2.1)

3 1
 
2 9



21
(2.1) det

 a b c d


Cofactor of 3 is
 ad
 bc
8

Chapter 2: Determinants

Minus place sign det



1 5
1 0


By (2.1)

1 0
 
1 5
 
5
Cofactor of 9 is det


1 5

2 0



By (2.1)

1 0
 
2 5



10
Cofactor of 7 is

Minus place sign det

 
1

1
2 1


    
2
  

By (2.1)
1
Cofactor of

2 is det



1 5
9 7



By (2.1)

1 7
 
9 5
  
52
Cofactor of 1 (the 1 on the bottom row of the given matrix) is

Minus place sign det


1 5
3 7


 
By (2.1)

1 7
 
3 5
 
8
Cofactor of the last entry 0 is det


1

1
3 9



By (2.1)

1 9
 
3
  

12
Hence by collecting these together and placing them in the corresponding position gives the cofactor matrix:
C



7

14 21
5 10 1




52 8 12 
As stated above, we use the cofactor matrix to find the inverse of an invertible matrix.
Definition (2.9). Let A be a square matrix then the matrix consisting of the cofactors of each entry in A is called the cofactor matrix and is normally denoted by C . The transpose of this cofactor matrix is called the adjoint of A and is denoted by
 
. That is adj
  
C
T
Remember we discussed the transpose of a matrix in the last chapter and it means swapping the rows and columns around.
Example 8
Find the adjoint of the matrix A given in Example 7 above.
Solution
We have already done all the hard work in evaluating the cofactor matrix C above. The adjoint is the transpose of this matrix C : adj
  
C
T



7 5

52
14 10 8


 21 1 12 




Because C



 

7

14 21
5 10 1


52 8 12 




A4 Inverse of a Matrix
9

Chapter 2: Determinants 10
The remainder of this section is very demanding because you are required to understand the proofs provided. It is going to be challenging to follow the remaining proofs but proving results gives you a better understanding of the concepts involved.
If you are struggling to understand the chain of arguments in the proof then come back and go over the proof a second time. It is not necessary that you understand every detail on the first reading. Good luck with your journey on this difficult terrain.
Proposition (2.10). If a square matrix A consists of two identical rows then det

0 .
What does this proposition mean?
Means if a matrix has two rows which are the same then the determinant of the matrix is 0.
Proof .
To be proved in the next section.
Proposition (2.11). Let A be a n by n square matrix. If C jk a jk
for k

1, 2, 3, and n then
denotes the cofactor of the entry a C j 1
 a C j 2
 a C j 3
  a C jn
 det
 0 if i
 j if i
 j
Proof .
How do we prove this result?
We consider the two cases i
 j and i
 j then show the required result in each case.
Case 1: Let i
 j then by the formula for determinant
(2.8) det
   a C i 1 i 1
 a C i 2 i 2
 a C i 3 i 3
  a C in in we have a C i 1 i 1
 a C i 2 i 2
 a C i 3 i 3
Case 2: Consider the case when i
 j :
  a C in in
 det
 
Let A * be the matrix obtained from matrix A by copying the entries of the ith row into the jth row of matrix A . That is the matrix A * is the matrix A but with jth row being identical to the ith row: ith row jth row
We have
A
 






 a
11 a
12 a i 1 a i a a j 1 j 2 det
  
0 a a n 1 n 2
. Why?
2 a
1 n a in a jn a nn








and A *
 






 a
11 a
12 a a i 1 i 1 a a i i 2
2 a a n 1 n 2 a
1 n a in a in a nn







 ith row =jth row
Because we have two identical rows , i and j, in A *
(2.10) the determinant is zero, that is det
  
0 .
therefore by the previous Proposition
Therefore if we expand along the jth row in the Right Hand matrix A *
0
 det
   a C i 1
* j 1
 a C i 2
* j 2
 a C i 3
* j 3
  a C in
* jn
we have
(†) where C * is the cofactor of entry jk a jk
 a ik
in the matrix A * .
Consider the cofactor C j 1
which is the place sign multiplied by the determinant of the remaining matrix after deleting the row and column containing the entry a j 1
in the Left Hand

Chapter 2: Determinants 11 matrix A . Similarly the cofactor C * j 1
is the place sign multiplied by the determinant of the remaining matrix after deleting the row and column containing the entry a
 a j 1 i 1
in the Right
Hand matrix A * . We have
C j 1
  j

1
1 det







 a j 1 a
12 a i 2 a n 2 a
1 n a in a nn








and C * j 1
  j

1
1 det jth row







 a i 1 a
12 a i 2 a n 2 a
1 n a in a nn








What can you conclude about C j 1
and C * ? j 1
C * j 1

C because the cofactor is made up of the same entries of matrix A and A * j 1
. In both cases you delete the jth row and first column. Similarly we have C * j 2

C j 2
, C * j 3

C j 3
, and C * jn

C jn
. Substituting these, C * j 1

C , j 1
C * j 2

C j 2
, and C * jn

C jn
, into (†) gives
0
 det
  
 a C i 1
* j 1
 a C i 2
* j 2
 a C i 3 a C i 1 j 1
 a C i 2 j 2
 a C i 3
Hence in the case where i
 j we have j 3
* j 3
  a C in
* jn
  a C in jn a C i 1 j 1
 a C i 2 j 2
 a C i 3 j 3
  a C in jn

0 which is our required result.
■
Proposition (2.12). Let A be a n n square matrix. Then
A adj
   det
 
Proof .
Writing out the entries of the general n by n matrix A and the transpose of the corresponding cofactors of each entry gives
A ith row
 






 a a a
21 22 a
11 i 1 a a
12 i 2 a n 1 a n 2 a
1 n a
2 n a in a nn








and adj


C
11
C
21
C C
12 22

 C
1 n
C
2 n
C j 1
C j 2
C jn jth column
C n 1
C n 2
C nn





Consider the ij entry of the Left Hand matrix multiplication in A adj
 
. How do we evaluate the ij entry in the matrix multiplication A adj
Remember for matrix multiplication it is row times column. So the ij entry of A adj
 
is the ith row times the jth column. For an ij entry we have

Chapter 2: Determinants 12
A adj  ij
 a C i 1 j 1
 a C i 2 j 2
 a C i 3 j 3
  a C in jn
 det
0 if i
 j if i
 j

By the above
 Proposition (2.11)


Repeating this for each ij entry and writing out the matrix A adj
 
means we have when i
 j , along the leading diagonal , and 0 everywhere else in the matrix A adj det
 
 
:
A adj



 det
0
0 det
0
0
0
0 




Taking Out
Common Factor det




1 0
0 1
0 det 
0
0 



 det
 
0 1

I
Hence we have our result, A adj
   det
 
.
■
Proposition (2.13). Let A be a square matrix. If det

0 then we have
A

1  det
1
  adj
Proof .
This follows from the previous proposition (2.12). Since det

0 we can divide the above formula given in Proposition (2.12) A adj
   det
 
by det
 
A


 det
1
  adj
 




I
Remember from the first chapter we know that A is invertible (non-singular) matrix if
AA

1 
I where A

1
is unique. Therefore A

1  det
1
  adj provided det

0 .
■
What does Proposition (2.13) mean?
The inverse of an invertible matrix A is given by the formula A

1  det
1
  adj . To determine the inverse of a matrix you need to find the cofactors and the determinant of the given matrix. What is the point of finding the inverse of a matrix?
As discussed in the last chapter, we need the inverse to solve linear system of equations, which will be discussed later in this chapter.

Chapter 2: Determinants
Example 9
Find the inverse of the matrix given in Example 7 which is
A


1

1 5
3 9 7




2 1 0 
Solution
We need to find A

1 which is given by the above Proposition: A

1  det
1
  adj
What is
Remember
  is the cofactor matrix transposed and was found in Example 8 above: adj
  
C
T



7 5

52
14 10 8


21 1 12
We only need to find det
 
. Expanding along the bottom row of the given matrix A because it contains a 0: det
 
2 det



1 5
9 7



1det


1 5
3 7



0
2

7 45
 
7 15
 
112
Substituting these into formula (2.13) gives




Because det

Expanding along this row

1

1 5 det

3 9 7


2 1 0







A

1  det
1
  adj

1
112




7 5

52
14 10 8


21 1 12
Normally we would find the determinant first and then the adjoint of the matrix because if the determinant is zero then the matrix is non-invertible (singular).
Check that the matrix found in Example 9 is indeed the inverse of A . How?
Check the matrix multiplication
 
1
I


1 0 0
0 1 0


. If you think this is a tedious task
0 0 1 use MATLAB.
Later in this chapter we use the inverse matrix to solve linear system of equations.
SUMMARY
The determinant and inverse of a 2 2 matrix A
 a b c d


is given respectively by
(2.1) det
 ad
 bc
(2.2) A

1  det
1
 

 d
 b
 c a


provided det

0
13

Chapter 2: Determinants
The minor denoted by M ij of an entry a ij
is the determinant of the remaining matrix after deleting the entries in the ith row and jth column containing that entry.
The cofactor of an entry is the minor multiplied by a place sign. The place sign for an entry a ij
is given by
  i
 j
.
The cofactor matrix of a given matrix consists of cofactor of each entry in its corresponding position.
If we expand along the ith row of a matrix A , then the formula for determinant is
(2.8) det
   a C i 1 i 1
 a C i 2 i 2
 a C i 3 i 3
  a C in in
The adjoint of a matrix A is the cofactor matrix transposed and it is denoted by
 
.
The inverse of a square matrix is defined as
(2.13) A

1  det
1
  adj provided det

0 .
14