2. Matrices and Matrix Operations
2.1 Matrices, addition, subtraction and multiplication by a number
2.1.1 Matrices
A matrix is a rectangular array of numbers. It is what we frequently call a table of
numbers.
Example 2.1.1. (a continuation of Example 1.3.1) An electronics company makes two
types of circuit boards for computers, namely ethernet cards and sound cards. Each of
these boards requires a certain number of resistors, capacitors and transistors as follows
resistors
capacitors
transistors
ethernet card
5
2
3
sound card
7
3
5
This table of numbers naturally forms what is called a 32 matrix. Let’s denote it by A.
5 7
A =  2 3 
3 5
It is called 32 matrix since it has three rows and two columns.
(5, 7)
(2, 3)
(3, 5)
5
2
 
3
is the first row
is the second row
is the third row
is the first column
7
3
 
2
is the second column
The individual elements of a matrix are referred to by pairs of subscripts, i.e.
Aij = the element in row i and column j of A.
In the example above A21 = 2. In fact, all six elements of the matrix above are labeled as
follows.
5 7
 A11 A12 


A A 
2
3
A = 
 =  21 22 
3 5
 A31 A32 
2.1 - 1
The rows of A can be regarded as row vectors. Let
Ai,● = row i of A.
In the example above A2,● = (2, 3). The columns of A can be regarded as column vectors.
Let
A●,j = column j of A.
7
In the example above A●,2 =  3 .
2
A column vector can be regarded as a n1 matrix where n is the number of components
4
in the vector. For example,  1  is a 31 matrix. A row vector can be regarded as a 1n
2
matrix where n is again the number of components in the vector. For example,
(-2, 3, 0, 6) is a 14 matrix.
The transpose of a matrix A is denoted by AT and obtained by making the rows of A the
columns of the transpose and the columns of A the rows of the transpose. Another way of
putting this is the element in row i and column j of the transpose is the element in row j
and column i of A, i.e. (AT)ij = Aji. The transpose of A is denoted by AT. If A is a mn
matrix then AT is a nm matrix. For example,
5 7
A =  2 3 
3 5

5 2 3
AT =  7 3 5 
AT has the same data as A, only the way of referring to it is different.
2.1.2 Addition, subtraction and multiplication by a number
One adds and subtracts matrices by adding and subtracting corresponding components.
In order to do this they have to have the same number of rows and columns. One
multiplies a matrix by a number c by multiplying each component by c. One forms the
negative of a matrix A by negating the components of A. For 32 matrices this looks like
 a11 a12 
 b11 b12 
A =  a21 a22 
B =  b21 b22 
 a31 a32 
 b31 b32 
a
+b
a
11
11
12+b12 

A + B =  a21+b21 a22+b22 
 a31+b31 a32+b32 
ca
 11 ca12 
cA =  ca21 ca22 
 ca31 ca32 

 a11-b11 a12-b12 
A - B =  a21-b21 a22-b22 
 a31-b31 a32-b32 
-a
 11 -a12 
- A =  -a21 -a22 
 -a31 -a32 
2.1 - 2
For example
5 7
A =  2 3 
2 -1
B =  4 -6  
5+2 7+(-1)
7 6
A + B =  2+4 3+(-6)  =  6 -3 
35 37   15 21 
3A = 
 32 33  =  6 9 
5-2 7-(-1)
3 8
A - B =  2-4 3-(-6)  =  -2 9 
-5 -7
- A =  -2 -3 
Thus (A+B)ij = Aij + Bij, (A-B)ij = Aij - Bij, (-A)ij = - (Aij) and (cA)ij = c(Aij) for i = 1, …, m
and j = 1, …, n where m is the number of rows in A and B and n is the number of columns.
0 0
A zero matrix is a matrix all of whose components are 0, e.g. A =  0 0  is the 32 zero
0 0
matrix.
The same algebraic properties that hold for the vector operations also hold for the matrix
operations. Here A, B and C are matrices with the same dimensions and a and b are
numbers. The first eight properties say that the set of matrices with a specified number of
rows and columns is a vector space.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
A+B=B+A
A + (B + C) = (A + B) + C
A+0=A
A + (-A) = 0
(a + b)A = aA + bA
a(A + B) = aA + aB
a (bA) = (ab) A
1A = A
It follows from Proposition 1 in section 1.2 that the other algebraic properties that hold in
a vector space hold for matrices, e.g. a0 = 0, 0A = 0, (-1)A = - A, A + (B – A) = B, (A + B) = (- A) + (- B).
Also, Proposition 2 in section 1.2 also holds for matrices, i.e.
(9)
(10)
(11)
(12)
(13)
(AT)T = A
(A + B)T = AT + BT
(A - B)T = AT - BT
(cA)T = cAT
(-A)T = -(AT)
2.1 - 3