Chapter 2
Matrices
Matrices provide an orderly way of
arranging values or functions to enhance
the analysis of systems in a systematic
manner. Their use is very important in
simplifying large arrays of equations and
in determining their solutions. In
particular, computer algorithms for
manipulating large arrays are simplified
greatly by matrix notation and operations.
1
A particular reason for introducing
matrices at this point is that MATLAB is
heavily matrix oriented. One does not
have to be an expert in matrix theory to
utilize the powerful features of MATLAB,
but it is very helpful to understand some
of the terminology and manipulations. This
chapter will deal with those basic elements
of matrix theory that can be used for that
purpose. The term linear algebra is often
used to represent the general theory of
matrices and the associated algebraic
operations.
2
General Form of a Matrix of Size m x
n with m Rows and n Columns
 a11 a12 a13 ....a1n 
 a a a ....a 
21 22
23
2n 

A
:
: 


 am1 am 2 am3 ...amn 
3
Square Matrix
If m = n, the matrix is said to be square.
In this case, the matrix could be
designated as an m x m matrix.
A m,m
4
Vectors
A matrix having only one row is called a
row matrix. A matrix having only one
column is called a column matrix. A
matrix of either form is called a vector.
The row matrix is called a row vector and
the column matrix is called a column
vector.
5
Scalars
A 1 x 1 matrix is called a scalar and is
the form of most variables considered
in simple algebraic forms.
6
Transpose
The transpose of a matrix A is denoted as
A’ and is obtained by interchanging the
rows and columns. Thus, if A has a size of
m x n, A’ will have a size of n x m. If the
transpose operation is applied twice, the
original matrix is restored.
7
Example 2-1. Determine the size of
the matrix shown below.
 2 -3 5
A

-1 4 6
The matrix has 2 rows and 3 columns. Its
size is 2 x 3.
8
Example 2-2. Determine the size of
the matrix shown below.
2 1 


B  7 -4 
3 1 
The matrix has 3 rows and 2 columns. Its
size is 3 x 2.
9
Example 2-3. Determine the size of
the matrix shown below.
 2 -1 3


C  4 6 1 
-5 2 1
The matrix has 3 rows and 3 columns. Its
size is 3 x 3. It is a square matrix.
10
Example 2-4. Express the integer values
of time from 0 to 5 s as a row vector.
t  [0 1 2 3 4 5]
The size of the vector is 1 x 6.
11
Example 2-5. Express the variables
x1, x2, and x3 as a column vector.
 x1 


x   x2 
 x3 
12
Example 2-6. Determine the
transpose of the matrix A below.
 2 -3 5
A

-1 4 6 
 2 -1 


A'  -3 4 
 5 6 
13
Addition and Subtraction of Matrices
Matrices can be added together or
subtracted from each other if and only if
they are of the same size. Corresponding
elements are added or subtracted.
Cm,n = Am,n ± Bm,n
14
Multiplication of Two Matrices
Two matrices can be multiplied together
only if the number of columns of the first
matrix is equal to the number of rows of
the second matrix. This means that
AB  BA
15
Multiplication of Two Matrices
(Continuation)
The number of rows in the product matrix
is equal to the number of rows in the first
matrix and the number of columns in the
product matrix is equal to the number of
columns in the second matrix.
A m,n Bn ,k  Cm,k
16
Multiplication of Two Matrices
(Continuation)
An element in the product matrix is
obtained by summing successive products
of elements in the row of the first with
elements of the column of the second.
n
cij   air brj
r 1
17
Division of Matrices ?
There is no such thing as division of
matrices. However, matrix inversion can
be viewed in some sense as a procedure
similar to division. This process will be
considered later.
18
Example 2-7. Determine C = A + B for
the matrices shown below.
 3 2 1
A

-4 5 6 
 4 9 x
B

 6 -3 y 
7 11 1+x 
C

 2 2 6+y 
19
Example 2-8. Determine D = A - B for
the matrices shown below.
 3 2 1
A

-4 5 6 
 4 9 x
B

 6 -3 y 
-1 -7 1-x 
D

-10 8 6-y 
20
Example 2-9. For the matrices of
Examples 2-1 and 2-2, determine
possible orders of multiplication.
 2 -3 5
A

-1 4 6
2 1 


B  7 -4 
3 1 
AB  A 2,3B 3,2  C2,2
BA  B 3,2 A 2,3  D3,3
21
Example 2-10. For the matrices of
Examples 2-1 and 2-2, determine
C=AB.
A = A 2,3
 2 -3 5


-1 4 6 
B = B 3,2
2 1 
 2 -3 5 

C
7
-4



-1
4
6

 3 1 


2 1 


 7 -4 
3 1 
22
Example 2-10. Continuation.
c11  (2)(2)  (3)(7)  (5)(3)  4  21  15  2
c12  (2)(1)  (3)(4)  (5)(1)  2  12  5  19
c21  (1)(2)  (4)(7)  (6)(3)  2  28  18  44
c22  (1)(1)  (4)(4)  (6)(1)  1  16  6  11
 2 19 
C

 44 -11
23
Example 2-11. For the matrices of
Examples 2-1 and 2-2, determine
D=BA.
A = A 2,3
 2 -3 5


-1 4 6 
B = B 3,2
2 1 


 7 -4 
3 1 
2 1 
 3 -2 16
 2 -3 5 



D = BA  7 -4 
 18 -37 11 

-1 4 6

3 1 
 5 -5 21
24
Determinants
The determinant of a matrix A can be
determined only for a square matrix. It is
a scalar value. Various representations
are shown as follows:
det(A)
A

25
Determinant of 2 x 2 Matrix
a11 a12 
A

a21 a22 
det( A)  a11a22  a12 a21
26
Determinants of Higher-Order
For determinants of matrices of higher
order than 2 x 2, the process can become
tedious. There are many “tricks”, but
some are useful only when the matrix has
simple numbers. The text provides a
procedure based on minors and cofactors,
but since the ultimate goal is to use
MATLAB, that procedure will not be
covered on these slides. Instead, formulas
for the 3 x 3 case will be provided.
27
Determinant of 3 x 3 Matrix
 a11 a12

A   a21 a22
 a31 a32
a13 

a23 
a33 
det( A)  a11 (a22 a33  a23a32 )
 a12 (a21a33  a23a31 )
 a13 (a21a32  a22 a31 )
28
Singular Matrix
If det(A)=0, the matrix is said to be
singular. If the matrix represents the
coefficients of a set of simultaneous
equations, it means that the equations
are not independent of each other and
cannot be solved uniquely.
29
Example 2-13. Determine the
determinant of the matrix shown below.
 3 2
A

-4 5
3 2
det( A ) 
 (3)(5)  (2)(4)
-4 5
 15  8  23
30
Examples 2-13, 2-14, and 2-15
These 3 examples involve computations
concerning the minors and cofactors of a
3 x 3 matrix and will not covered in this
presentation. The results that follow these
computations will be considered in a later
example.
31
Identity Matrix
1 0 0....0 
0 1 0....0 


I  0 0 1....0 


:
:


0 0 0 ....1
32
Inverse Matrix
The inverse of a matrix A is denoted by
A-1 and is defined by the equation that
follows.
1
1
AA  A A  I
33
Inverse of a 2 x 2 Matrix
 a22 -a12 
1


-a21 a11 
 a11 a12 

a a  
det( A)
 21 22 
-a12 
 a22
 det( A) det( A) 


a11 
 -a21
 det( A) det( A) 


34
Example 2-16. Determine the
inverse of the matrix A below.
 2 3
A 
 4 5
det( A)  (2)(5)  (3)(4)  2
 5 -3
1


-4 2   2.5 1.5
 2 3

1
A 




-1
2
 4 5
 2
35
Example 2-17. The inverse of A
below is developed in the text.
 1 2 -1


A  -1 1 3
 3 2 1
-0.25 -0.2 0.35


1
A  0.5 0.2 -0.1 
-0.25 0.2 0.15 
36
Simultaneous Linear Equations
a11 x1  a12 x2  ....a1m xm  b1
a21 x1  a22 x2  ....a2 m xm  b2
:
:
:
am1 x1  am 2 x2  ....amm xm  bm
37
Matrix Form of Simultaneous
Linear Equations
 a11 a12 ....a1m   x1  b1 
 a a ....a   x  b 
2m   2 
2 
 21 22


:
:  :  : 

   
 am1 am 2 ....amm   xm  bm 
38
Define variables as follows:
 a11 a12 ....a1m 
 a a ....a 
21 22
2m 

A
:
: 


 am1 am 2 ....amm 
 x1 
b1 
x 
b 
2 
2 


x
b
: 
: 
 


 xm 
bm 
39
Matrix Solution Development
Ax  b
A Ax  A b
-1
-1
-1
A A=I
Ix = x
40
The general form and the final
solution follow.
Ax = b
-1
x=A b
41
Example 2-18. Use matrices to solve
the simultaneous equations below.
x1  2 x2  x3  8
 x1  x2  3x3  7
3x1  2 x2  x3  4
42
Example 2-18. Continuation.
 1 2 -1   x1   8
-1 1 3  x    7 

 2  
 3 2 1  x3   4 
Ax = b
43
Example 2-18. Continuation.
-1
x=A b
 x1  -0.25 -0.2 0.35  -8 






x   x2   0.5 0.2 -0.1   7
 x3  -0.25 0.2 0.15   4
 2


x  -3 
 4 
44
Transformation of Linear Variables
y1  b11 x1  b12 x2
y2  b21 x1  b22 x2
 y1  b11 b12   x1 

 y  b b   x 
 2   21 22   2 
y = Bx
45
Transformation Continuation
z1  a11 y1  a12 y2
z2  a21 y1  a22 y2
 z1   a11 a12   y1 
 z   a a   y 
 2   21 22   2 
z = Ay
46
Transformation Continuation
z = Ay
y = Bx
z = ABx = Cx
C = AB
47
Example 2-19. For the system of
equations provided below, determine
the z values in terms of the x values.
y1  2 x1  3x2
y2  4 x1  2 x2
z1  5 y1  2 y2
z2  4 y1  3 y2
48
Example 2-19. Continuation.
 y1   2 -3  x1 
 y    4 -2  x 
 2
 2 
 z1  5 -2  y1 
 z   4 3   y 
 2
 2 
49
Example 2-19. Continuation.
 z1  5 -2  2 -3  x1 
 z    4 3   4 -2  x 

 2
 2 
 z1   2 -11  x1 
 z    20 -18  x 
 2
 2 
50