Matrix Arithmetic and the Transpose of a Matrix
A matrix is an array of numbers. A row matrix,
 x1
x2
...
x n  is a 1Xn matrix.This is sometimes
 x1 
 
x2
called a vector. A column matrix,   is nX1 and is also called a vector.
  
 
 xn 
If A is mxn, the entry in row I, column j is a ij .
 1
3
 2
4
Example: For A  
For two matrices, A, B ,
5 
 find the size of the matrix and find a12 and a 23 .
 5
A  B if and only if a ij  bij for every I and j and the matrices are of the
same size.
In the context of matrices, ordinary numbers are called scalars. To multiply a matrix by a scalar, multiply
every entry by the scalar.
Example: For A as in the example above,
 1
2 A  2
 2
3
4
5   2

 5  4
6
8
10 
.
 10 
Addition: If A and B have the same size then A  B is the matrix formed by adding entries in the
Same position. That is, the row i, column j entry of A  B is a ij  bij .
 9

Example: A  8

 2
5 

6

 7 
 6

B  2

  3
4 

1 , find the matrix 3 A  2 B .

 2 
Matrices can be used to store data. Suppose two people, John and Mary each invest in stocks 1, 2 and 3.
The tables show the amount each person has in each stock.
John
Mary
S1
1000
2500
S2
3400
1700
S3
5000
6500
S1
S2
S3
John
1000
3400
5000
Mary
2500
1700
6500
The 2nd table is the transpose of the first. To form the transpose, we made the 1st row become the 1st
column and the 2nd row became the 2nd column.
We form the transpose of a matrix the same way, row i becomes column i.
T
T
The transpose of the matrix A is written A . If A is mxn then A is nxm.
 1

2
Example: C 

  2
I
3
5
7
8
3
5
6
T

0 Find C .

6 