Elementary row operations are used to transform a system of linear equations into a new system that has the same solutions as the original one (i.e., into an equivalent system).
There are three elementary operations:
multiplying an equation by a non-zero constant;
adding a multiple of an equation to another equation;
interchanging two equations.
A system of
linear equations in
unknowns is
written in
matrix form
as
where:
is the
matrix of coefficients;
is the
vector of unknowns;
is the
vector of constants.
The rows of the system are the
equations
where:
is the
-th
row of
(it contains the coefficients of the
-th
equation);
is the
-th
entry of
.
The first elementary operation we consider is the multiplication of an
equation by a constant
.
If the
-th
equation is the one being multiplied, then we substitute the
equation
with
the
equation
The original matrix of coefficients and vector of
constantsbecome
so
that the new system
is
The same result can be achieved as follows:
take the
identity matrix
;
multiply the
-th
row of
by
and denote the transformed matrix thus obtained by
:
pre-multiply both sides of the matrix form of the system by
:
It can be easily verified
that
In the lecture on
Equivalent
systems, we have proved that if
is invertible, then the new
system is equivalent to the original one.
But the matrix
above is invertible
(full-rank) because its rows
are linearly independent
(none of them can be written as a
linear combination of the
others).
Thus, multiplying an equation by a non-zero constant gives an equivalent system.
Example
Consider the system of two equations in three
unknownsthat
can be written in matrix form as
where
Multiplying
the second equation by
,
we obtain the equivalent
system
that
can be written in matrix form
as
where
The
same result can be achieved by 1) taking the
identity
matrix
2)
multiplying its second row by
so as to obtain the
matrix
and
3) pre-multiplying
and
by
:
The second elementary row operation we consider is the addition of a multiple of an equation to another equation.
Suppose we want to add
times the
-th
equation to the
-th
equation. Then we substitute the
equation
with
the
equation
The original matrix of coefficients and vector of
constants
become
so
that the new system
is
The same result can be achieved as follows:
take the
identity matrix
;
add
times the
-th
row of
to the
-th
row of
,
and denote the transformed matrix thus obtained by
:
pre-multiply both sides of the matrix equation by
:
As before, we have
thatand
the new system is equivalent to the original one because
is invertible (none of its rows can be written as a linear combination of the
others).
In other words, we obtain an equivalent system by adding a multiple of one row to another row.
Example
Consider the system of three equations in three
unknownsthat
can be written in matrix form as
where
Let
us add the second equation multiplied by
to the third one. We obtain the equivalent
system
that
can be written in matrix form
as
where
The
same result can be achieved by 1) taking the
identity
matrix
2)
multiplying its second row by
and adding it to the third one so as to obtain the
matrix
and
3) pre-multiplying
and
by
:
The third elementary row operation we consider is the interchange of two equations.
We switch the
-th
equation
with
the
-th
equation
The original matrix of coefficients and vector of
constantsbecome
so
that the new system
is
The same result can be obtained as follows:
take the
identity matrix
;
switch the
-th
row of
with the
-th
row
(
in the original matrix), and denote the new matrix by
:
pre-multiply both sides of the system by
:
As for the previous elementary operations, we have
thatand
the new system is equivalent to the original one because
is invertible (the rows of
are the same of
,
but in a different order; they form the
standard basis of the space of
vectors).
To sum up, we obtain an equivalent system by interchanging two rows (two equations) of the system.
Example
Consider the system of three equations in three
unknownsthat
can be written in matrix form as
where
Let
us switch the first equation with the third one. We get the equivalent
system
that
can be represented in matrix form
as
where
The
interchange of equations can also be performed by 1) starting from the
identity
matrix
2)
switching the first row with the third
one
and
3) pre-multiplying
and
by
:
Below you can find some exercises with explained solutions.
Suppose that
is
a system of
equations in
unknowns.
What is the matrix
that allows us to interchange the second equation with the fourth (when the
system is pre-multiplied by
)?
The matrix
is obtained by interchanging the rows of the
identity matrix:
Suppose that we have a system of
equations in
unknowns.
What is the matrix
that allows us to multiply the second equation by
?
The matrix
is obtained by multiplying by
the second row of the
identity matrix:
Suppose that we have a system of
equations in
unknowns.
What is the matrix
that allows us to add the first equation to the second?
The matrix
is obtained by adding the first row of the
identity matrix to the second:
Please cite as:
Taboga, Marco (2021). "Elementary row operations", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/elementary-row-operations.
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