Welcome to a lesson on elementary matrices.
The goals of this video are to define an elementary matrix
and also to determine if a given matrix
is an elementary matrix.
An elementary matrix E is an n by n matrix,
so it must be square, it can be obtained
from the identity matrix by one elementary row operation.
So little e is an elementary row operation.
We can express an elementary matrix as big E
equals little e of I.
All elementary matrices are invertible,
meaning their inverse exists and the inverse
of an elementary matrix is also
an elementary matrix.
And for review here are some identity matrices.
Notice how the main diagonal from the upper left
to the lower right are ones
and all of their elements are zero.
And the elementary row operation consists of the following.
We can interchange two rows,
multiply a row by a nonzero constant,
or add or subtract a multiple of one row to another row.
Let's look at some examples.
To determine if this given matrix
is an elementary matrix, we want to determine
if we can perform one row operation
on the identity matrix to transform
it into this matrix here.
Notice how the only change is
this element here is now three.
So we want to figure out if we
can perform one row operation on this matrix
to transform it into this matrix.
Well if we were to replace row two
with three times row one plus row two,
notice how this would be three plus zero would be three
and this element here would just
be three times zero plus one,
so it would still be one.
And since we obtained this matrix here
by performing just one row operation
on this identity matrix the answer is yes
this is an elementary matrix.
Let's look at several more examples.
Again we're given this matrix,
so we want to start with a three by three identity matrix
and see if we can obtain the given matrix
by performing one row operation.
Notice the only change is this element here
is negative three, so if we were to replace
row two with negative three times row two
notice how we would obtain the given matrix
and therefore again the answer is yes
this is an elementary matrix.
Next this is the given matrix.
So we start with the three by three identity matrix
and see if we can obtain the given matrix
by performing one row operation on the identity matrix.
Again notice the only change is this element here
is zero, so we might be thinking
we can just replace row two with zero times row two.
But multiplying a row by zero
is not a elementary row operation
and therefore the answer is no
this is not an elementary matrix.
Now we're given this three by three matrix.
Again starting with the three by three identity matrix
we want to see if we can perform one row operation
in this identity matrix to obtain the given matrix.
Well notice how this matrix here has been changed two ways.
This one is now four and this one is now a negative one.
So the only way to obtain the given matrix
would be to replace row two with four time row two
and to replace row three with negative one times row three.
Well this would be two elementary row operations
and for this to be an elementary matrix
we can only perform one row operation
therefore again the answer is no
this is not an elementary matrix.
Let's look at two more examples.
This is our given matrix, so we start with
the three by three identity matrix
and see if we can perform one row operation
to obtain the given matrix.
Well notice in the given matrix or this matrix here
these two elements have changed and so have these.
But if we interchange row two and row three
we would obtain the given matrix.
Therefore the answer is yes
the given matrix is an elementary matrix.
And for our last example,
notice how we're given a two by three matrix
but an elementary matrix must be
a square matrix so right away we know
the answer is no this is not an elementary matrix.
There's no way we can start with a two by two
or three by three identity matrix
perform one row operation to obtain a two by three matrix.
It's just not possible.
Okay, I hope you found these examples helpful.