- [Instructor] So let me just write a general
two-by-two matrix A.
So let's just say its elements are a, b, c, and d.
Now, from previous videos, we have learned
how to find the inverse of our matrix A,
the formula that we went over.
The inverse of our matrix A is going to be equal to
one over the determinant of our matrix A
times what is often called the adjoint of our matrix A.
And that's you are swapping the a and the d.
So you have a d here and an a here.
You're making the b and the c negative.
And this works for a two-by-two when you do a,
when you go to a three-by-three case,
it gets a little bit more complicated.
So we're gonna have a negative b here
and a negative c here.
Or another way to think about it,
the determinant of a two-by-two matrix
is going to be one over ad minus bc,
ad minus bc,
and then you multiply that times the adjoint.
So that is fancy word,
that's a fancy word for it.
So we have d, negative b, negative c, and a.
So that is all review.
If it's not, I encourage you to review it
in other places on Khan Academy.
But what we're going to ask ourselves in this video is,
is there a situation where a matrix is not invertible?
So pause this video,
and see if you can figure out a scenario
or a general set of scenarios
where a matrix is not going to be invertible.
So one way to think about a matrix that is not invertible
is that, for that matrix,
we would somehow not be able to make this calculation.
Now, no matter what a, b, c, and d are,
we can always find an adjoint matrix.
But what's a situation where one over the determinant
is we're not going to be able to calculate it?
Well, a situation where the determinant,
the determinant of our matrix A is equal to zero,
or I could write the determinant of our matrix A
is going to be equal to zero.
In that situation, we're not going to be able to calculate
one over the determinant because that is undefined.
And so we say,
we say A
is not invertible,
invertible,
if and only if, I'll do this two-way arrow right over here,
our determinant for our matrix A is equal to zero.
Because if our determinant is equal to zero,
we're going to have to divide by zero
when we find the inverse.
Now, let's explore what that
would look like a little bit more.
If our determinant is going to be equal to zero,
that means that this expression right over here
is going to be equal to zero.
So that means that ad minus bc is equal to zero.
We can add bc to both sides.
That means that a times d is equal to b times c.
And then we can algebraically manipulate
this a little bit more if we divided both sides by b and d.
So let me just do that.
If I divide both sides by b and d,
then the d's cancel out here, and the b's cancel out here.
So this could be a over b
is equal to c over d.
So we have a situation where the,
you could think of it as the ratio between a and b
is equivalent to the ratio between c and d.
You could also think about it the other way.
You could divide,
let's start with that original equation, ad is equal to bc.
If you divided both sides by
c and d,
so you divide both sides by cd,
what you're going to be left with is that a over c
is going to be equal to b over d.
And so the ratio between a and c
is equal to the ratio between b and d.
So all of those situations,
you're going to have a noninvertible matrix
because the determinant is going to be equal to zero.
And so let's look at a few examples.
I want you to tell me whether the following matrices
are invertible or not.
So let's say I have the matrix five, one,
three, two.
Is this an invertible matrix?
Pause this video, and try to figure it out.
So the determinant of this matrix is going
to be equal to five times two, 10,
minus three times one,
minus three, which is equal to seven.
And since this is not equal to zero,
we know that this matrix is invertible.
We can find an inverse.
Now, what about this matrix?
So 10, 20,
and 250 and 500,
is this matrix invertible?
The determinant here,
of 10, 20,
250, and 500,
this is going to be equal to 10 times 500, which is 5,000,
minus 20 times 250, which is 5,000, which is equal to zero.
And once again, if you wanted to find its inverse,
you would take your adjoint matrix
and multiply it times one over the determinant.
But one over zero is not defined, so this is not invertible.