Elementary Matrix Operations
Elementary matrix operations play an important role in many matrix
algebra applications, such as
finding the inverse of a matrix
and
solving simultaneous linear equations.
Elementary Operations
There are three kinds of elementary matrix operations.
 Interchange two rows (or columns).
 Multiply each element in a row (or column) by a nonzero number.
 Multiply a row (or column) by a nonzero number and add the
result to another row (or column).
When these operations are performed on rows, they are called
elementary row operations; and when they are performed on
columns, they are called elementary column operations.
Elementary Operation Notation
In many references, you will encounter a compact notation to describe
elementary operations. That notation is shown below.
Operation description 
Notation 
Row operations 
1. Interchange rows i and j 
R_{i} <> R_{j} 
2. Multiply row i by s, where s ≠ 0 
sR_{i} > R_{i} 
3. Add s times row i to row j 
sR_{i} + R_{j} > R_{j} 
Column operations 
1. Interchange columns i and j 
C_{i} <> C_{j} 
2. Multiply column i by s, where s ≠ 0 
sC_{i} > C_{i} 
3. Add s times column i to column j 
sC_{i} + C_{j} > C_{j} 
Elementary Operators
Each type of elementary operation may be performed by matrix multiplication,
using square matrices called
elementary operators.
For example, suppose you want to interchange rows 1 and 2 of Matrix
A. To accomplish this, you could premultiply
A by E to produce
B, as shown below.
R_{1} <> R_{2} = 

0 + 2 
0 + 4 
0 + 6 

0 + 1 
0 + 3 
0 + 5 

Here, E is an elementary operator. It operates on
A to produce the desired interchanged rows in
B. What we would like to know, of course,
is how to find E. Read on.
How to Perform Elementary Row Operations
To perform an elementary row operation on a
A, an r x c matrix, take the following
steps.
 To find E, the
elementary row operator,
apply the operation to an r x r
identity matrix.
 To carry out the elementary row operation, premultiply
A by E.
We illustrate this process below for each of the three types of elementary
row operations.
Interchange two rows. Suppose we want to interchange
the second and third rows of A, a 3 x 2 matrix. To
create the elementary row operator E, we interchange
the second and third rows of the identity matrix
I_{3}.
Then, to interchange the second and third rows
of A,
we premultiply A by E, as
shown below.
R_{2} <> R_{3} = 

1*0 + 0*2 + 0*4 
1*1 + 0*3 + 0*5 

0*0 + 0*2 + 1*4 
0*1 + 0*3 + 1*5 
0*0 + 1*2 + 0*4 
0*1 + 1*3 + 0*5 



Multiply a row by a number. Suppose we want to
multiply each element in the second row of Matrix A
by 7. Assume A is a 2 x 3 matrix. To
create the elementary row operator E, we multiply each
element in the second row of the identity matrix
I_{2} by 7.
Then, to multiply each element in
the second row of A by 7,
we premultiply A by E.
7R_{2} > R_{2} = 

1*0 + 0*3 
1*1 + 0*4 
1*2 + 0*5 

0*0 + 7*3 
0*1 + 7*4 
0*2 + 7*5 

Multiply a row and add it to another row.
Assume A is a 2 x 2 matrix. Suppose we want to
multiply each element in the first row of A
by 3; and we want to add that result to the second row of
A. For this
operation, creating the elementary row operator is a twostep process.
First, we multiply each
element in the first row of the identity matrix
I_{2} by 3. Next, we add the result of
that multiplication to the second row of I_{2}
to produce E.
Then, to multiply each element in
the first row of A by 3 and add that result to the
second row,
we premultiply A by E.
3R_{1} + R_{2} > R_{2} = 




E 

A 

3R_{1} + R_{2} > R_{2} = 

1*0 + 0*2 
1*1 + 0*3 

3*0 + 1*2 
3*1 + 1*3 

3R_{1} + R_{2} > R_{2} = 

How to Perform Elementary Column Operations
To perform an elementary column operation on
A, an r x c matrix, take the following
steps.
 To find E, the
elementary column operator,
apply the operation to an c x c
identity matrix.
 To carry out the elementary column operation, postmultiply
A by E.
Let's work through an elementary column operation to illustrate the
process. For example, suppose we want to interchange
the first and second columns of A, a 3 x 2 matrix. To
create the elementary column operator E, we interchange
the first and second columns of the identity matrix
I_{2}.
Then, to interchange the first and second columns
of A,
we postmultiply A by E, as
shown below.
C_{1} <> C_{2} = 

0*0 + 1*1 
0*1 + 1*0 

2*0 + 3*1 
2*1 + 3*0 
4*0 + 5*1 
4*1 + 5*0 

Note that the process for performing an elementary column operation on an
r x c matrix is very similar to the process for performing
an elementary row operation. The main differences are:
 To operate on the r x c matrix A,
the row operator E is created from an r x r
identity matrix;
whereas the column operator E is created from an
c x c identity matrix.
 To perform a row operation,
A is premultiplied by E; whereas
to perform a column operation, A is postmultiplied
by E.
Test Your Understanding
Problem 1
Assume that A is a 4 x 3 matrix. Suppose you want to
multiply each element in the second column of matrix A
by 9. Find the elementary column operator E.
Solution
To find the elementary column operator E, we multiply each
element in the second column of the identity matrix
I_{3} by 9.