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What is a matrix?
A rectangular array of numbers (we will concentrate on
real numbers). A nxm matrix has ‘n’ rows and ‘m’
columns
M11 M12 M13 M14  First row
M3x4

 M21 M22
M31 M32
M23
M33

M24 
M34 
First Second Third Fourth
column column column column
M12
Row number
Column number
Second row
Third row
What is a vector?
A vector is an array of ‘n’ numbers
A row vector of length ‘n’ is a 1xn matrix
a1
a2
a3
a4 
A column vector of length ‘m’ is a mx1 matrix
 a1 
a 
 2
a 3 
Special matrices
Zero matrix: A matrix all of whose entries are zero
03 x 4
0 0 0 0
 0 0 0 0
0 0 0 0
Identity matrix: A square matrix which has ‘1’ s on the
diagonal and zeros everywhere else.
I3x 3
1 0 0


 0 1 0
0 0 1
Matrix operations
Equality of matrices
If A and B are two matrices of the same size,
then they are “equal” if each and every entry of one
matrix equals the corresponding entry of the other.
 1 2 4
a b c 




A   3 0 7  B   d e f 
 9 1 5
 g h i 
a  1, b  2, c  4,
A  B  d  3, e  0, f  7,
g  9,
h  1,
i  5.
Matrix operations
Addition of two
matrices
If A and B are two matrices of the same size,
then the sum of the matrices is a matrix C=A+B whose
entries are the sums of the corresponding entries of A and
B
 1 2 4
  1 3 10 
A   3 0 7 B    3 1 0 
 9 1 5
 1 0 6 
 0 5 14 


C  A  B   6 1 7 
 10 1 11
Addition of of matrices
Matrix operations
Properties
Properties of matrix addition:
1. Matrix addition is commutative (order of
addition does not matter)
AB  B A
2. Matrix addition is associative
A  B  C  A  B  C
3. Addition of the zero matrix
A0  0A  A
Matrix operations
Multiplication by a
scalar
If A is a matrix and c is a scalar, then the product cA is a
matrix whose entries are obtained by multiplying each of
the entries of A by c
1

A   3
 9
 3

cA    9
 27
2 4

0 7 c  3
1 5
6 12 

0 21
3 15 
Matrix operations
Multiplication by a
scalar
Special case
If A is a matrix and c =-1 is a scalar, then the product
(-1)A =-A is a matrix whose entries are obtained by
multiplying each of the entries of A by -1
 1 2 4


A    3 0 7
 9 1 5
1

cA  -A   3
  9
c  1
 2  4

0  7
 1  5
Matrix operations
Subtraction
If A and B are two square matrices of the same
size, then A-B is defined as the sum A+(-1)B
 1 2 4
  1 3 10 




A    3 0 7 B    3 1 0 
 9 1 5
 1 0 6 
2  1  6


C  A  B  0  1 7 
8 1  1
Note that A - A  0 and 0 - A  -A
Special
operations
Transpose
If A is a mxn matrix, then the transpose of A is
the nxm matrix whose first column is the first
row of A, whose second column is the second
column of A and so on.
 1 2 4
1  3 9 




T
A    3 0 7   A   2 0 1
 9 1 5
4 7 5
Special
operations
Transpose
If A is a square matrix (mxm), it is called
symmetric if
AA
T
Matrix operations
Scalar (dot) product of
two vectors
If a and b are two vectors of the same size
 a1 
 b1 




a  a 2 ; b  b2 
a3 
b3 
The scalar (dot) product of a and b is a scalar
obtained by adding the products of
corresponding entries of the two vectors
a  b  a1b1  a2b2  a3b3 
Matrix operations
Matrix multiplication
For a product to be defined, the number of columns
of A must be equal to the number of rows of B.
A
mxr
B
rxn
inside
outside
=
AB
mxn
Matrix operations
Matrix multiplication
If A is a mxr matrix and B is a rxn matrix, then the
product C=AB is a mxn matrix whose entries are
obtained as follows. The entry corresponding to row ‘i’
and column ‘j’ of C is the dot product of the vectors
formed by the row ‘i’ of A and column ‘j’ of B
A 3x3
C3x2
 1 2 4
 1



  3 0 7 B 3x2    3
 9 1 5
 1
 3 5 




 AB   10  9 notice 
  7 28 

3

1
0
1   1



2    3  3
4  1 
Matrix operations
Multiplication of
matrices
Properties
Properties of matrix multiplication:
1. Matrix multiplication is noncommutative
(order of addition does matter)
AB  BA in general
 It may be that the product AB exists but BA
does not (e.g. in the previous example
C=AB is a 3x2 matrix, but BA does not
exist)
 Even if the product exists, the products AB
and BA are not generally the same
Matrix operations
Multiplication of
matrices
Properties
2. Matrix multiplication is associative
ABC   ABC
3. Distributive law
AB  C  AB  AC
B  CA  BA  CA
4. Multiplication by identity matrix
AI  A; IA  A
5. Multiplication by zero matrix A0  0; 0A  0
T
T T
6.
AB
B A
Matrix operations
Miscellaneous
properties
1. If A , B and C are square matrices of the
same size, and A  0 then AB  AC
does not necessarily mean that B  C
2. AB  0 does not necessarily imply that
either A or B is zero
Inverse of a
matrix
Definition
If A is any square matrix and B is another
square matrix satisfying the conditions
AB  BA  I
Then
(a)The matrix A is called invertible, and
(b) the matrix B is the inverse of A and is
denoted as A-1.
The inverse of a matrix is unique
Inverse of a
matrix
Uniqueness
The inverse of a matrix is unique
Assume that B and C both are inverses of A
AB  BA  I
AC  CA  I
(BA)C  IC  C
B(AC)  BI  B
BC
Hence a matrix cannot have two or more
inverses.
Some properties
Inverse of a
matrix
Property 1: If A is any invertible square
matrix the inverse of its inverse is the matrix A
itself
-1 1
A 
A
Property 2: If A is any invertible square
matrix and k is any scalar then
k A 
1
1 -1
 A
k
Properties
Inverse of a
matrix
Property 3: If A and B are invertible square
matrices then
1
1 -1
(AB) AB 
1
AB
I
B A
Premultiplying both sides by A-1
A (AB) AB   A 1
1
-1
A ABAB
1
-1
 A 1
BAB   A 1
1
Premultiplying both sides by B-1
AB 1  B 1A 1
What is a determinant?
The determinant of a square matrix is a number
obtained in a specific manner from the matrix.
For a 1x1 matrix:
A  a11 ; det( A)  a11
For a 2x2 matrix:
a11 a12 
A
; det( A)  a11a 22  a12a 21

a 21 a 22 
Product along red arrow minus product along blue arrow
Example 1
Consider the matrix
1 3
A

5
7


Notice (1) A matrix is an array of numbers
(2) A matrix is enclosed by square brackets
1 3
det( A) 
 1  7  3  5  8
5 7
Notice (1) The determinant of a matrix is a number
(2) The symbol for the determinant of a matrix is
a pair of parallel lines
Computation of larger matrices is more difficult
Duplicate column method for 3x3 matrix
For ONLY a 3x3 matrix write down the first two
columns after the third column
a11 a12
A  a21 a 22
a31 a 32
a13 
a 23 
a 33 
 a11 a12 a13  a11 a12
a

 21 a 22 a 23  a 21 a 22
a 31 a 32 a 33  a 31 a 32
Sum of products along red arrow
minus sum of products along blue arrow
det(A)  a11a 22a 33  a12a 23a 31  a13a 21a 32
 a13a 22a 31  a11a 23a 32  a12a 21a 33
This technique works only for 3x3 matrices
Example
2 4 - 3
A  1 0 4

 2 - 1 2




2
1

2
0 -8
 3 2

4 1

 1 2  2
4
0
8
0
4
0
1
32
3
Sum of red terms = 0 + 32 + 3 = 35
Sum of blue terms = 0 – 8 + 8 = 0
Determinant of matrix A= det(A) = 35 – 0 = 35
Finding determinant using inspection
Special case. If two rows or two columns are proportional
(i.e. multiples of each other), then the determinant of the
matrix is zero
2
3
7
2
8
4 0
2 7 8
because rows 1 and 3 are proportional to each other
If the determinant of a matrix is zero, it is called a
singular matrix
Cofactor method
What is a cofactor?
If A is a square matrix
a11 a12
A  a21 a 22
a31 a 32
a13 
a 23 
a 33 
The minor, Mij, of entry aij is the determinant of the submatrix
that remains after the ith row and jth column are deleted from A.
The cofactor of entry aij is Cij=(-1)(i+j) Mij
M12 
a 21 a 23
a 31 a 33
 a 21a 33  a 23a 31 C12   M12  
a 21 a 23
a 31 a 33
What is a cofactor?
Sign of cofactor





-





Find the minor and cofactor of a33
2 4 - 3
A  1 0 4

 2 - 1 2




Minor
2 4
M 33 
 2  0  4  1  4
1 0
Cofactor C  ( 1)(33) M  M  4
33
33
33
Cofactor method of obtaining the
determinant of a matrix
The determinant of a n x n matrix A can be computed by
multiplying ALL the entries in ANY row (or column) by
their cofactors and adding the resulting products. That is,
for each 1  i  n and 1  j  n
Cofactor expansion along the jth column
det( A)  a1jC1j  a2jC2j    anjCnj
Cofactor expansion along the ith row
det( A)  ai1Ci1  ai2Ci2    ainCin
Example: evaluate det(A) for:
A=
1 0
2
-3
3 4
0
1
-1 5
2
-2
0 1
1
3
0
1
4
det(A)=(1) 5 2 -2
1 1 3
3 4 0
- (-3) -1 5 2
0
1
1
det(A) = a11C11 +a12C12 + a13C13 +a14C14
3
0
1
- (0) -1 2 -2
0
1
3
+2
3
4
-1
5 -2
0
1
= (1)(35)-0+(2)(62)-(-3)(13)=198
1
3
Example : evaluate
1
det(A)= 1
5 -3
0
2
3 -1 2
By a cofactor along the third column
det(A)=a13C13 +a23C23+a33C33
4 1
-3*
(-1)
det(A)=
3
0
-1
+2*(-1)5
1
5
3
-1
= det(A)= -3(-1-0)+2(-1)5(-1-15)+2(0-5)=25
+2*(-1)6
1
5
1
0
Quadratic form
The scalar
Ud kd
T
d  vector
k  square matrix
Is known as a quadratic form
If U>0: Matrix k is known as positive definite
If U≥0: Matrix k is known as positive semidefinite
Quadratic form
Let
Then
 d1 
 k11 k12 
d  k 

d 2 
k 21 k 22 
Symmetric
matrix
 k11 k12   d1 
U  d k d  d1 d 2 
 

k12 k 22  d 2 
 k11d1  k12 d 2 
 d1 d 2 

k12 d1  k 22 d 2 
 d1 (k11d1  k12 d 2 )  d 2 (k12 d1  k 22 d 2 )
T
 k11d1  2k12 d1 d 2  k 22 d 2
2
2
Differentiation of quadratic form
Differentiate U wrt d1
U
 2k11d1  2k12 d 2
d1
Differentiate U wrt d2
U
 2k12 d1  2k 22 d 2
d 2
Differentiation of quadratic form
Hence
 U 
 k11
U  d1 

 2

 d  U 
k12
 d 2 
2k d
k12   d1 
 

k 22  d 2 