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Matrix Multiplication
To Multiply matrix A by matrix B:
•Multiply each Row in matrix A by each Column in matrix B
•Multiply corresponding entries and then add the resulting products
R1
R1
R2
1 2
A
3 4
(1)(-1) + (2)(3)
(1)(1) + (2)(-2)
B
C1
C2
C3
1
1
2
3
2 3
(1)(2) + (2)(3)
5
-3
8
9
-5
18
=
A.B =
(3)(-1) + (4)(3)
(3)(1) + (4)(-2)
(3)(2) + (4)(3)
We had:
2 elements or
2 columns
2 rows
1 2
A
3 4
,
2 elements or
2 rows
3 columns
B
1
3
1
Result:
2 rows by 3 columns
2
2 3
and
A.B 
5
3
8
9  5 18
A: has 2 rows, 2 columns or 2 x 2
B: has 2 rows, 3 columns or 2 x 3
By multiplying Rows from the first matrix by Columns in the second matrix:
• The result will have: number of rows of A and number of columns of B.
The result AB has 2 rows and 3 columns or 2 x 3.
• The number of elements in per row of A, must be equal to the number of
elements in per column in B, Or:
Number of columns in the A = Number of Rows in B
2
=
2
For the following matrices, using the multiplication of Row by Column :
A
a)
b)
1 1
5
2 1 1
,
2
B  1
0
,
2
C 1
0
1
2
4
Which of the following multiplication is possible
If it is possible, find the dimension of the resulting matrix
A.B:
a) the number of elements per row in A (3 elements, 3 columns)
the number of element per column in B (3 elements, 3 rows).
b) The resulting matrix will be 2 row by 1 columns or 2 x 1
3
A.B 
5
A.C:
a) the number of elements per row in A (3 elements, 3 columns)
the number of element per column in C (3 elements, 3 rows).
A.C 
b) The resulting matrix will be 2 rows by 2 columns or 2 x 2
B.C:
C.A:
3
21
5 4
a) the number of elements per row in B (1 elements, 1 columns)
the number of element per column in C (3 elements, 3 rows).
B.C is Not Possible
a) the number of elements per row in C (2 elements, 3 columns)
the number of element per column in A (2 elements, 2 rows).
0 1 11
C. A.  5 3 3
8 4 4
b) The resulting matrix will be 3 rows by 3 columns or 3 x 3
The following example will be helpful in Markov Chain section (Section 9.2).
If: A 
1
1
2
0
find A2, A3, A4 and A5
A2  A. A 
1 1 1 1 1 1
.

2 0 2 0
2 2
A3  A2 . A 
1 1 1 1 3 1
.

2 2 2 0
2 2
1 1 1 1
1 3
A  A .A 
.

2 2 2 2 6 2
4
2
2
A5  A2 . A3 
1 1 3 1
5 1
.

2 2 2 2 2 6