Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture we introduced matrices
and saw how augmented matrices allow us to represent systems of linear equations
in a compact easy-to-manipulate form.
In this lecture, we will introduce three matrix operations
called "elementary row operations"
which are used to transform an augmented matrix
into a matrix representing a simpler system of equations
with identical solutions to the original system.
Two systems of equations which have different forms but identical solutions
are said to be "equivalent".
For example, both of these systems of equations
describe three planes which intersect at a single point.
Although the planes in the first system are different from the planes in the second system
the point of intersection of the planes is the same in both systems.
Since both systems of planes intersect at the same point
they both have the same solution and are therefore equivalent.
We have seen that solving a system of linear equations using the elimination method
involves performing a series of operations which simplify the equations
by eliminating variables.
This produces an equivalent system of equations
which indicate the solution set of the system.
Likewise, solving a system of linear equations using matrices
involves performing a series of elementary row operations which simplify the matrix
and eliminate variables in the corresponding equations.
We saw in the previous lecture
that when solving a system of linear equations using matrices
the first step is to create an augmented matrix
which is a shorthand representation of the system of equations.
Each row in the augmented matrix corresponds to one equation in the system.
As we will see in this lecture, the second step is to transform this matrix
into one corresponding to an equivalent system of simpler equations.
We do this by using elementary row operations.
There are three types of elementary row operations
which we will refer to as "swap", "scale" and "pivot".
The first type of elementary row operation is the "swap" operation.
This operation allows us to exchange any two rows of the augmented matrix.
For example, using three swap operations we could exchange rows one and two
rows one and three
and rows two and three.
When using elementary row operations to transform a matrix
it is helpful to have a record of the sequence of operations used
so that later we can remember how the transformed matrix was produced.
Many textbooks use a method of notating row operations which we will use as well.
To notate that we have used a swap operation to swap row one and row two
we write "R subscript one"
and "R subscript two"
separated by a double-headed arrow.
Since we also swapped rows one and three and rows two and three
we want to note that we had performed these swap operations as well.
Swapping rows changes the form but not the solutions of the system
since the order in which the equations of a system are listed
has no effect on the system's solutions.
Since none of these operations change the solutions of the system
we will have produced an equivalent system of equations.
The second type of elementary row operation is the "scale" operation.
This operation allows us to multiply any row of the augmented matrix
by a non-zero constant.
For example, using the scale operation
we could multiply all the elements in row one by the constant "two".
The result would then replace all the entries in row one.
To notate that we have used a scale operation to multiply row one by the constant two
we write "two R subscript one"
followed by an arrow and "R subscript one"
indicating that two times row one
replaces row one.
So let's perform a couple of scale operations and see how the matrix changes.
We will start by using a scale operation to multiply row one by the constant "two"
and then perform a second scale operation to multiply row three by "negative one-third".
Since multiplying both sides of an equation by a non-zero constant
does not affect its solutions
the scale operation always produces an equivalent system of equations.
The third type of elementary row operation is the "pivot" operation.
This operation allows us to add a multiple of any row to another row
replacing that row.
For example, using a pivot operation
we could multiply row two by the constant "negative three"
and add that multiple of row two to row one
replacing row one with the result.
To notate that we have used a pivot operation
to add "negative three" times row two to row one
we write "R subscript one"
minus "three R subscript two"
followed by an arrow and "R subscript one"
indicating that negative three times row two was added to row one
with the sum replacing row one.
So let's perform this pivot operation.
We start by multiplying row two by negative three
and add that multiple of row two to row one
replacing row one with the result.
Once again, the pivot operation changes the form but not the solutions of the system.
The system of equations represented by the transformed matrix
may describe different planes than those contained in the original system
but the intersection points of those planes and the corresponding solutions
will remain unchanged.
The transformed system will therefore be equivalent
to the original system of equations.
In this last example, we transformed the augmented matrix
into a new one representing an equivalent system.
In this process, the first row of the augmented matrix was changed
and so the corresponding equations were different.
Since in the transformed equation, the x-coefficient had become zero
the variable "x" was eliminated from the equation.
In the next lecture we will introduce a method called "Gaussian elimination"
which uses the three elementary row operations to transform and simplify augmented matrices
eliminating variables in the corresponding system of equations.