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Section 4.1 Matrix Addition and Scalar Multiplication
Matrix, Dimension and Entries
A m x n matrix, A is a rectangular array of real numbers with m
rows and n columns. We refer to m and n as the dimensions of
the matrix. The numbers that appear in the matrix are called its
entries. We customarily use capital letters A , B , C ,….. for the
names of matrices
 3 7 52 
A

-6 12 37
is a 2 x 3 matrix because it has two rows and
three columns.
 3
-6
B
39

19






is a 4 x 1 matrix because it has four rows and
one column.
The entries of A are 3, 7, 52, -6, 12, and 37
The entries of B are 3, -6, 39, and 19.
We refer to the entries by their row and column designation. So for
matrix A the second row and first column designation would be
listed as a21 and in this case that value would be a21  6 .
Generically for any position in A we would have aij where i would
designate the row and j would designate the column.
Example 1
Find the dimension of the given matrix and identify the given entry
 3 13 
-6 22 
 a32
A
39 17 


19 8 
Two matrices are considered equal if they have the same
dimensions and the corresponding entries are equal.
Example 2
Solve for x, y, z, and w
 x-y
12. 
 y-w
x-z  0


w  0
0 
6 
Row Matrix, Column Matrix, and Square Matrix
A matrix with a single row is called a row matrix or row vector.
G   13 5 88 this matrix is a 1 x 3
A matrix with a single column is called a column matrix or column
vector.
 1 
-23 
 this matrix is a 4 x 1
D
 16 


 17 
A matrix with the same number of rows as columns is called a
square matrix
 7 52
H 
 this square matrix is a 2 x 2
12
37


Matrix Addition and Subtraction
Two matrices can be added ( or subtracted) if and only if they have
the same dimensions. To add (or subtract) two matrices of the
same dimensions, we add (or subtract) the corresponding entries.
More formally, if A and B are m x n matrices, then A + B and A - B
are the m x n matrices whose entries are given by:
 A  Bij  Aij  Bij
 A  Bij  Aij  Bij
Scalar Multiplication
If A is an m x n matrix and c is a real number, then c A is the m x n
matrix obtained by multiplying all the entries of A by c . Generally
lowercase letters c, d , e are used to denote scalars. So the ij th entry
of cA is given by
cAij  cAij 
Transposition
If A is an m x n matrix, then its transpose is the n x m matrix,
obtained by writing its rows as columns, so the ith row of the
original matrix becomes the ith column of the transpose. We denote
the transpose of the matrix A by A T
3 - 6 
 3 7 52 
T
7 12 
A

A



-6 12 37
52 37 
Properties of Transposition
If A and B are m x n matrices, then the following hold:
 A  B T  AT  BT
cAT  cAT 
A 
T T
A
Example 3
 0 -1 
 0.25 - 1 
 1 -1 
0.5 , C   1 1 
Given A   1 0  , B   0






3 
- 1
- 1 2 
- 1 - 1
14. A  C
17*. 2 A  C
20*. 2 AT  4 BT