This document provides an overview of matrix algebra concepts including: - Definitions of matrices, operations on matrices like addition, subtraction, scalar multiplication, and matrix multiplication. - Transpose of a matrix. - Determinants of matrices and how they are calculated. - Inverse of a matrix, how it is defined, and methods for calculating the inverse including using determinants and Gaussian elimination. - Examples are provided to illustrate key concepts and calculations.

1. Fin500J Mathematical Foundations in Finance
Topic 1: Matrix Algebra
Philip H. Dybvig
Reference: Mathematics for Economists, Carl Simon and Lawrence Blume, Chapter 8 Chapter 9
and Chapter 16
Slides designed by Yajun Wang
1
Fall 2010 Olin Business School
Fin500J Topic 1

2. Outline
 Definition of a Matrix
 Operations of Matrices
 Determinants
 Inverse of a Matrix
 Linear System
 Matrix Definiteness
Fall 2010 Olin Business School 2
Fin500J Topic 1

3. Matrix (Basic Definitions)
 
ij
kn
k
n
n
A
a
a
a
a
a
a
,
,
,
,
,
,
1
2
21
1
11























A
An k × n matrix A is a rectangular array of numbers with k rows and n
columns. (Rows are horizontal and columns are vertical.) The numbers k and n
are the dimensions of A. The numbers in the matrix are called its entries. The
entry in row i and column j is called aij .
3
Fall 2010 Olin Business School
Fin500J Topic 1

4. Operations with Matrices (Sum, Difference)
Sum, Difference
If A and B have the same dimensions, then their sum, A + B, is obtained by
adding corresponding entries. In symbols, (A + B)ij = aij + bij . If A and B
have the same dimensions, then their difference, A − B, is obtained by
subtracting corresponding entries. In symbols, (A - B)ij = aij - bij .
Example:
4
Fall 2010 Olin Business School
A
A 



























0
A
all
for
Then,
zero.
all
are
entries
whose
0
matrix
The
1
12
12
8
4
2
1
5
6
7
0
1
0
7
6
1
4
3
Fin500J Topic 1

5. Operations with Matrices (Scalar Multiple)
Scalar Multiple
If A is a matrix and r is a number (sometimes called a scalar in this
context), then the scalar multiple, rA, is obtained by multiplying every
entry in A by r. In symbols, (rA)ij = raij .
Example:
5
Fall 2010 Olin Business School

















0
14
12
2
8
6
0
7
6
1
4
3
2
Fin500J Topic 1

6. Operations with Matrices (Product)
Product
If A has dimensions k × m and B has dimensions m × n, then the product
AB is defined, and has dimensions k × n. The entry (AB)ij is obtained
by multiplying row i of A by column j of B, which is done by multiplying
corresponding entries together and then adding the results i.e.,
6
Fall 2010 Olin Business School
B.
IB
B,
matrix
m
n
any
for
and
A
AI
A,
matrix
n
m
any
for
1
0
0
0
1
0
0
0
1
I
matrix
Identity
.
Example
.
...
)
...
( 2
2
1
1
2
1
2
1








































































n
n
mj
im
j
i
j
i
mj
j
j
im
i
i
fD
eB
fC
eA
dD
cB
dC
cA
bD
aB
bC
aA
D
C
B
A
f
e
d
c
b
a
b
a
b
a
b
a
b
b
b
a
a
a








Fin500J Topic 1

7. Laws of Matrix Algebra
 The matrix addition, subtraction, scalar multiplication and matrix
multiplication, have the following properties.
Fall 2010 Olin Business School 7
BC.
AC
B)C
AC, (A
AB
C)
A(B
A
B
B
A
A(BC).
C, (AB)C
B)
(A
C)
(B
A















:
Laws
ve
Distributi
:
Addition
for
Law
e
Commutativ
:
Laws
e
Associativ
Fin500J Topic 1

8. Operations with Matrices (Transpose)
Transpose
The transpose, AT , of a matrix A is the matrix obtained from A by
writing its rows as columns. If A is an k×n matrix and B = AT then
B is the n×k matrix with bij = aji. If AT=A, then A is symmetric.
Example:
8
Fall 2010 Olin Business School
,
C
D
(CD)
rA
(rA)
A,
)
(A
,
B
A
B)
(A
,
B
A
B)
(A
a
a
a
a
a
a
a
a
a
a
a
a
T
T
T
T
T
T
T
T
T
T
T
T
T
T































Then,
matrix.
n
m
an
be
D
and
matrix
m
k
a
be
C
Let
scalar.
a
is
r
and
n
k
are
B
and
A
where
:
verify
easy to
it
It
23
13
22
12
21
11
23
22
21
13
12
11
Fin500J Topic 1

9. Determinants
 Determinant is a scalar
 Defined for a square matrix
 Is the sum of selected products of the elements of the matrix, each
product being multiplied by +1 or -1
11 12 1
21 22 2
1 1
1 2
det( ) ( 1) ( 1)
n
n n
n i j i j
ij ij ij ij
j i
n n nn
a a a
a a a
A a M a M
a a a
 
 
    
 
9
Fall 2010 Olin Business School
• Mij=det(Aij), Aij is the (n-1)×(n-1)
submatrix obtained by deleting row
i and column j from A.
Fin500J Topic 1

10. Determinants
 The determinant of a 3 ×3 matrix is
11 12 13
22 23 21 23 21 22
1 1 1 2 1 3
21 22 23 11 12 13
31 32
32 33 31 33
31 32 33
( 1) ( 1) ( 1)
a a a
a a a a a a
a a a a a a
a a
a a a a
a a a
  
     
 Example
10
1 1 1 2 1 3
1 2 3
5 6 4 6 4 5
4 5 6 1( 1) 2( 1) 3( 1)
8 10 7 10 7 8
7 8 10
50 48 2(40 42) 3(32 35) 3
  
     
       
10
bc
ad
d
c
b
a
A 


)
det(
 The determinant of a 2 ×2 matrix A is
• In Matlab: det(A) = det(A)
Fall 2010 Olin Business School
Fin500J Topic 1

11. Inverse of a Matrix
 Definition. If A is a square matrix, i.e., A has dimensions n×n. Matrix
A is nonsingular or invertible if there exists a matrix B such that
AB=BA=In. For example.
 Common notation for the inverse of a matrix A is A-1
 If A is an invertible matrix, then (AT)-1 = (A-1)T
 The inverse matrix A-1 is unique when it exists.
 If A is invertible, A-1 is also invertible  A is the inverse matrix of A-1.
(A-1)-1=A.
• In Matlab: A-1 = inv(A)
• Matrix division:
A/B = AB-1
11
Fall 2010 Olin Business School


















































 1
0
0
1
3
2
3
1
3
2
3
2
3
1
3
1
3
1
3
2
3
1
3
1
3
1
3
2
2
1
1
1
Fin500J Topic 1

12. Calculation of Inversion using Determinants
Def: For any n×n matrix A, let Cij denote the (i,j) th cofactor of A, that is, (-1)i+j
times the determinant of the submatrix obtained by deleting row i and column j
form A, i.e., Cij = (-1)i+j Mij . The n×n matrix whose (i,j)th entry is Cji, the (j,i)th
cofactor of A is called the adjoint of A and is written adj A.
thus
-1
Thm: Let A be a nonsingular matrix. Then,
1
A .
det
adj A
A

12
12
Fall 2010 Olin Business School
Fin500J Topic 1

13. Calculation of Inversion using Determinants
thus
Example: find the inverse of the matrix
Solve:
2 4 5
0 3 0
1 0 1
A
 
 
  
 
 
11 12 13
21 22 23
31 32 33
11 21 31
12 22 32
13 23 33
1
3 0 0 0 0 3
3, 0, 3,
0 1 1 1 1 0
4 5 2 5 2 4
4, 3, 4,
0 1 1 1 1 0
4 5 2 5 2 4
15, 0, 6,
3 0 0 0 0 3
det 9,
3 4 15
0 3 0 .
3 4 6
3
1
,
9
C C C
C C C
C C C
A
C C C
adjA C C C
C C C
So A
         
          
         
 
 
   
   
  
   
   

   
 
4 15
0 3 0 .
3 4 6
 
 
 

 
 

 
13
Using Determinants to find the
inverse of a matrix can be very
complicated. Gaussian elimination is
more efficient for high dimension matrix.
13
Fall 2010 Olin Business School
Fin500J Topic 1

14. Calculation of Inversion using Gaussian Elimination
14
 Elementary row operations:
o Interchange two rows of a matrix
o Change a row by adding to it a multiple of another row
o Multiply each element in a row by the same nonzero number
• To calculate the inverse of matrix A, we apply the elementary row
operations on the augmented matrix [A I] and reduce this matrix to the
form of [I B]
• The right half of this augmented matrix B is the inverse of A
14
Fall 2010 Olin Business School
Fin500J Topic 1

15. Calculation of inversion using Gaussian elimination
I is the identity matrix, and use Gaussian elimination to obtain a matrix of the form
The matrix


























1
0
0
,
,
0
1
0
,
,
0
0
1
,
,
]
[
1
2
21
1
11



nn
n
n
n
a
a
a
a
a
a
I
A
15
Fall 2010 Olin Business School






















nn
n
n
n
a
a
a
a
a
a
,
,
,
,
,
,
1
2
21
1
11
A

























nn
n
n
n
n
b
b
b
b
b
b
b
b
b
1
0
0
0
1
0
0
0
1
2
1
2
22
21
1
12
11


























nn
n
n
n
n
b
b
b
b
b
b
b
b
b
B
2
1
2
22
21
1
12
11
is then the matrix inverse of A
Fin500J Topic 1

16. Example
The matrix
1 1 1 |1 0 0
[ | ] 12 2 3 |0 1 0
3 4 1 |0 0 1
A I
 
 
 
 
 
 
16
1 1 1
12 2 3
3 4 1
 
 
 
 
 
 
A
is then the matrix inverse of A
1 1 1 | 1 0 0
0 10 15 | 12 1 0
0 0 3.5 | 4.2 0.1 1
 
 
  
 
 
 
 
3 1
1 0 0 | 0.4
35 7
2 3
0 1 0 | 0.6
35 7
1 2
0 0 1 | 1.2
35 7
 

 
 
 
 
 
 
 
 
 
 
(ii)+(-12)×(i), (iii)+(-3) ×(i), (iii)+(ii) ×(1/10)
3 1
0.4
35 7
2 3
0.6
35 7
1 2
1.2
35 7
 

 
 
 
 
 
 
 
 
 
 
16
Fall 2010 Olin Business School
Fin500J Topic 1

17. The system of linear equations
17
Fall 2010 Olin Business School
Systems of Equations in Matrix Form
11 1 12 2 13 3 1 1
21 1 22 2 23 3 2 2
1 1 2 2 3 3
n n
n n
k k k kn n k
a x a x a x a x b
a x a x a x a x b
a x a x a x a x b
    
    
    
can be rewritten as the matrix equation Ax=b, where
1 1
11 1
2 2
1
, , .
n
k kn
n k
x b
a a
x b
A x b
a a
x b
   
     
     
  
     
     
     
   
If an n×n matrix A is invertible, then it is nonsingular, and the
unique solution to the system of linear equations Ax=b is x=A-1b.
Fin500J Topic 1

18. Example: solve the linear system
1
-1
Matrix Inversion
4 1 2 x 4
5 2 1 ; X y ; b 4
1 0 3 z 3
6 -3 -3
1
A -14 10 6
6
-2 1 3
x 6 -3 -3 4
1
y -14 10 6 4
6
z -2 1 3 3
1 2; y 1 3; z 5 6
AX d
A
X A b
x


     
     
  
     
     
     

 
 
  
 
 
     
     

     
     
     
  
18
4 2 4
5 2 4
3 3
x y z
x y z
x z
  


  

  

Fall 2010 Olin Business School
• In Matlab
>>x=inv(A)*b
or
>> x=Ab
b
Fin500J Topic 1

19. 19
Fall 2010 Olin Business School
Matrix Operations in Matlab
>> A=[2 3; 1 1; 1 0]
A =
2 3
1 1
1 0
>> B1=[1 1; 0 1; 2 4]
B1 =
1 1
0 1
2 4
>> B2=[1 1 1; 1 0 2]
B2 =
1 1 1
1 0 2
>> A+B1
ans =
3 4
1 2
3 4
>> A-B1
ans =
1 2
1 0
-1 -4
>> A*B2
ans =
5 2 8
2 1 3
1 1 1
Sum
Difference
Product
Fin500J Topic 1

20. 20
Fall 2010 Olin Business School
Matrix Operations in Matlab
>> C=[1 1 1; 12 2 -3;
3 4 1]
C =
1 1 1
12 2 -3
3 4 1
>> C'
ans =
1 12 3
1 2 4
1 -3 1
>> det(C)
ans =
35
>> inv(C)
ans =
0.4000 0.0857 -0.1429
-0.6000 -0.0571 0.4286
1.2000 -0.0286 -0.2857
transpose
determinant
inverse
Fin500J Topic 1

21. Positive Definite Matrix
Fall 2010 Olin Business School 21
Fin500J Topic 1

22. Negative Definite, Positive Semidefinite, Negative
Semidefinite, Indefinite Matrix
Fall 2010 Olin Business School 22
Let A be an N×N symmetric matrix, then A is
• negative definite if and only if vTAv <0 for all v≠0 in RN
• positive semidefinite if and only if vTAv ≥0 for all v≠0, in RN
• negative semidefinite if and only if vTAv ≤0 for all v≠0, in
RN
• indefinite if and only if vTAv >0 for some v in RN and <0 for
other v in RN
Fin500J Topic 1