1.1.3 Matrix addition and matrix/vector multiplication
For the linear system of N equations for N unknowns:
a11x1 + a12x2 + … + a1NxN = b1
a21x1 + a22x2 + … + a2NxN = b2
:
:
aN1x1 + aN2x2 + … + aNNxN = bN
(1.1.3-1)
expressed in matrix/vector for as
Ax =b
(1.1.3-2)
We know that from ∫ 1.1.2 how to manipulate the N-dimensional real vectors
x,b,∈ RN.
 x1 
 x2 
 
x = : 
 
: 
 xn 
 b1 
 b2 
 
b = : 
 
: 
bn 
(1.1.3-3)
We write the matrix A as
a11
a21

A = :

:
an1
a12
a22
:
:
an2
a1n 
a2n 
... a2j ...

: :
:

: :
:

... anj ...
ann 
...
a1j ...
2nd row
aij = element of A in row #I and column #j.
jth column
If the number of columns (N) equals the number of rows (N), A is called a square
matrix.
To describe the size of a matrix with M rows and N columns, it is common to call it a
M “by” N or M x N matrix.
How do we manipulate matrices? First, look at some simple operations.
-multiplication of a M x N matrix A by a scalar c:
a11 a12 ... a1N 
a21 a22 ... a2N 


=
cA = c :
: :


: :

:
a M1 a M2 ... a MN 
ca11 ca12 ... ca1N 
ca21 ca22 ... ca2N 



:
: :


: :

:
ca M1 ca M2 ... ca MN 
(1.1.3-5)
-Addition of a M x N matrix A with a M x N matrix B:
a11 a12 ... a1N  b11 b12 ... b1N 
a21 a22 ... a2N  b21 b22 ... b2N 
 


 + :
 =
A + B = :
: :
: :
 


: :
: :
 :

:
a M1 a M2 ... a MN  b M1 b M2 ... b MN 

a11 + b11 a12 + b12 ... a1N + b1N

a21 + b a22 + b ... a2N + b
21
22
2N



:
(1.1.3-6)
: :


: :

:
a M1 + b M1 a M2 + b M2 ... a MN + b MN 


Note that A + B = B + A (1.1.3-7) and that two matrices can be added only if both the
number of rows and columns of each matrix are the same.
Other properties, such as c(A + B) = cA + cB are easily established.
-Multiplication of a N x N matrix A with an N-dimensional vector v :
If we are to have equivalence of notation between the set of linear algebraic equations
(1.1.3-1) and the matrix/vector equation (1.1.3-2), written explicitly below as:
a11
a21

:

:
an1
a12 a13 ...
a22 a23 ...
:
:
:
:
an2 an3 ...
a1n 
a2n 
: 

: 
ann 
 x1   b1 
 x2] b2 
   
:  = : 
   
:  : 
 xN  bn 
(1.1.3-9)
then the rule for multiplying (to be accurate, pre-multiplying) an N-dimensional
vector v by an N x N matrix A must be:
a11
a21

A v = :

:
an1
a12 a13 ...
a22 a23 ...
:
:
:
:
an2 an3 ...
a1n 
a2n 
: 

: 
ann 
 v1  a 11 v11 a 12 v 2 ... a 1N v N 
 v2] a v a v ... a v 
2N N 
   21 1 22 2

:  = :
: :

  
: :

:  :
 vN  a N1 v1 a N2 v 2 ... a NN v N 
(1.1.3-10)
We see that A v is also an N-dimensional vector, whose jth component (i.e. the value
in the jth row of A v ) is
(A v ) = aj1v1 + aj2v2 + … + ajNvN =
∑
N
k =1
a jk v k
(1.1.3-11)
This formula defines a summation of products across row #j of the matrix and down
the vector,
 

 

 ↓ 

→ a jk →  v k  ⇒ (A v)
j
 

 ↓ 

 

 

-Multiplication of an M x N matrix A with an N-dimensional vector v :
From the rule for A v just presented, it is clear that the number of columns of A must
equal the number of elements of v , but we can also define A v when M ≠ N.
a 11 a 12 a 13 ... a 1N 
a a

 21 22 a 23 ... a 2N 

A v = :
: :
:


: :
:
:

a M1 a M2 a M3 ... a MN 
 4444
1
4244444
3
M x N matrix A
 v1 
v 
 2
: 
 
: 
v N 
{
N -dimensional Vector v
a 11 v11 a 12 v 2 ... a 1N v N 
a v a v ... a v 
2N N 
 21 1 22 2

= :
: :


: :
:

a M1 v1 a M2 v 2 ... a MN v N 
 4444244443
1
M -dimensional vector
(1.1.3-12)
For example,
1 
1 2 3 4    30 
4 3 2 1  2 = 20 

 3   
11 12 13 14   130
 4
1
3

4

5
2
1
5
6
3
14 
1   

2   11 
2 =
6    32 
 3   
4    29
(1.1.3-13)
(1.1.3-14)
Note also that:
A(c v ) = cA v
A( v + w ) = A v + A w
(1.1.3-15)