Singular Matrices

Perhaps even more interesting
than finding the inverse of a

matrix is trying to determine
when an inverse of a matrix

doesn't exist. Or when
it's undefined.

And a square matrix for which
there is no inverse, of which

an inverse is undefined is
called a singular matrix.

So let's think about what a
singular matrix will look

like, and how that applies to
the different problems that

we've address using matrices.

So if I had the other 2
by 2, because that's

just a simpler example.

But it carries over into really
any size square matrix.

So let's take our
2 by 2 matrix.

And the elements are
a, b, c and d.

What's the inverse
of that matrix?

This hopefully is a bit of
second nature to you now.

It's 1 over the determinant of
a, times the adjoint of a.

And in this case, you just
switch these two terms. So you

have a d and an a.

And you make these two
terms negative.

So you have minus
c and minus b.

So my question to you is, what
will make this entire

expression undefined?

Well it doesn't matter what
numbers I have. If I have

numbers here that make a
defined, then I can obviously

swap them or make them negative,
and it won't change

this part of the expression.

But what would create a problem
is if we attempted to

divide by 0 here.

If the determinant of the
matrix A were undefined.

So A inverse is undefined, if
and only if-- and in math they

sometimes write it if with two
f's-- if and only if the

determinant of A
is equal to 0.

So the other way to view that
is, if a determinant of any

matrix is equal to 0, then
that matrix is a singular

matrix, and it has no inverse,
or the inverse is undefined.

So let's think about in
conceptual terms, at least the

two problems that we've looked
at, what a 0 determinant

means, and see if we can get a
little bit of intuition for

why there is no inverse.

So what is a 0 determinant?

In this case, what's a
determinant of this 2 by 2?

Well the determinant of
A is equal to what?

It's equal to ad minus bc.

So this matrix is singular, or
it has no inverse, if this

expression is equal to 0.

So let me write that
over here.

So if ad is equal to bc-- or we
can just manipulate things,

and we could say if a/b is equal
to c/d-- I just divided

both sides by b, and divided
both sides by d-- so if the

ratio of a:b is the same as the
ratio of c:d, then this

will have no inverse.

Or another way we could write
this expression, if a/c-- if I

divide both sides by c, and
divide both sides by d-- is

equal to b/d.

So another way that this would
be singular is if-- and it's

actually the same way.

If this is true, then
this is true.

These are the same.

Just a little bit of algebraic
manipulation.

But if the ratio of a:c is equal
to the ratio of b:d, and

you can think about why
that's the same thing.

The ratio of a:b being
the same thing as

the ratio of c:d.

But anyway, I don't want
to confuse you.

But let's think about how that
translates into some of the

problems that we looked at.

So let's say that we wanted to
look at the problem-- Let's

say that we had this matrix
representing the linear

equation problem.

Well, actually, this would
be either one.

So I have a, b, c, d times x,
y Is equal to two other

numbers that we haven't
used yet, e and f.

So if we have this matrix
equation representing the

linear equation problem, then
the linear equation problem

would be translated a times x
plus b times y is equal to e.

And c times x plus d times
y is equal to f.

And we would want to see where
these two intersect.

That would be the solution,
the vector

solution to this equation.

And so, just to get a visual
understanding of what these

two lines look like, let's
put it into the

slope y-intercept form.

So this would become what?

In this case, y is
equal to what?

y is equal to minus
a/b, x plus e/b.

I'm just skipping some steps.

But you subtract ax
from both sides.

And then divide both sides
by b, and you get that.

And then this equation, if you
put it in the same form, just

solve for y.

You get y is equal to minus
c/d x plus f/y.

So let's think about this.

I should probably change colors
because it looks too--

Let's think about what these two
equations would look like

if this holds.

And we said if this holds, then
we have no determinant,

and this becomes a singular
matrix, and it has no inverse.

And since it has no inverse, you
can't solve this equation

by multiplying both sides by
the inverse, because the

inverse doesn't exist.

So let's think about this.

If this is true, we have no
determinant, but what does

that mean intuitively in terms
of these equations?

Well if a/b is equal to c/d,
these two lines will have the

same slope.

They'll have the same slope.

So if these two expressions are
different, then what do we

know about them?

If two lines that have the
same slope and different

y-intercepts, they're parallel
to each other, and they will

never, ever intersect.

So let me draw that, just so you
get the-- this top line--

They don't have to be positive
numbers, but since this has a

negative, I'll draw it
as a negative slope.

So that's the first line.

And its y-intercept
will be e/b.

That's this line right here.

And then the second line-- let
me do it in another color-- I

don't know if it's going to be
above or below that line, but

it's going to be parallel.

It'll look something
like this.

And that line's y-intercept--
so that's this line-- that

line's y intercept is
going to be f/y.

So if e/b and f/y are different
terms, but both

lines have the same equation,
they're going to be parallel

and they'll never intersect.

So there actually would
be no solution.

If someone told you-- just the
traditional way that you've

done it, either through
substitution, or through

adding or subtracting the
linear equations-- you

wouldn't be able to find a
solution where these two

intersect, if a/b
is equal to c/d.

So one way to view the singular
matrix is that you

have parallel lines.

Well then you might say, hey
Sal, but these two lines would

intersect if e/b equaled f/y.

If this and this were the
same, then these would

actually be the identical
lines.

And not only would they
intersect, they would

intersect in an infinite
number of places.

But still you would have
no unique solution.

You'd have no one solution
to this equation.

It would be true at all
values of x and y.

So you can kind of view it when
you apply the matrices to

this problem.

The matrix is singular, if the
two lines that are being

represented are either parallel,
or they are the

exact same line.

They're parallel and not
intersecting at all.

Or they are the exact same line,
and they intersect at an

infinite number of points.

And so it kind of makes
sense that the A

inverse wasn't defined.

So let's think about this in
the context of the linear

combinations of vectors.

That's not what I wanted
to use to erase it.

So when we think of this problem
in terms of linear

combination of factors, we can
think of it like this.

That this is the same thing as
the vector ac times x plus the

vector bd times y, is equal
to the vector ef.

So let's think about
it a little bit.

We're saying, is there some
combination of the vector ac

and the vector bd that
equals the vector ef.

But we just said that if we have
no inverse here, we know

that because the determinant
is 0.

And if the determinant is 0,
then we know in this situation

that a/c must equal b/d.

So a/c is equal to b/d.

So what does that tell us?

Well let me draw it.

And maybe numbers would
be more helpful here.

But I think you'll get
the intuition.

I'll just draw the
first quadrant.

I'll just assume both vectors
are in the first quadrant.

Let me draw.

The vector ac.

Let's say that this is a.

Let me do it in a
different color.

So I'm gonna draw
the vector ac.

So if this is a, and this
is c, then the vector

ac looks like that.

Let me draw it.

I want to make this neat.

The vector ac is like that.

And then we have the arrow.

And what would the vector
bd look like?

Well the vector bd--
And I could draw

it arbitrarily someplace.

But we're assuming that there's
no derivative-- sorry,

no determinant.

Have I been saying derivative
the whole time?

I hope not.

Well, we're assuming
that there's no

determinant to this matrix.

So if there's no determinant,
we know that

a/c is equal to b/d.

Or another way to view it is
that c/d is equal to d/b.

But what that tells you is that
both of these vectors

kind of have the same slope.

So if they both start at point
0, they're going to go in the

same direction.

They might have a different
magnitude, but they're going

to go in the same direction.

So if this is point b, and this
is point d, vector bd is

going to be here.

And if that's not obvious to
you, think a little bit about

why these two vectors, if this
is true, are going to point in

the same direction.

So that vector is going to
essentially overlap.

It's going to have the same
direction as this vector, but

it's just going to have
a different magnitude.

It might have the
same magnitude.

So my question to you is, vector
ef, we don't know where

vector ef is.

Well let's pick some
arbitrary point.

Let's say that this is
e, and this is f.

So this is vector ef up there.

Let me do it in a
different color.

Vector ef, let's
say it's there.

So my question to you is, if
these two vectors are in the

same direction.

Maybe of different magnitude.

Is there any way that you can
add or subtract combinations

of these two vectors to
get to this vector?

Well no, you can scale these
vectors and add them.

And all you're going to do is
kind of move along this line.

You can get to any
other vector.

There's a multiple of one
of these vectors.

But because these are the exact
same direction, you

can't get to any vector that's
in a different direction.

So if this vector is in a
different direction, there's

no solution here.

If this vector just happened to
be in the same direction as

this, then there would be a
solution, where you could just

scale those.

Actually, there would be an
infinite number of solutions

in terms of x and y.

But if the vector is slightly
different, in terms of its

direction, then there
is no solution.

There is no combination of this
vector and this vector

that can add you
up to this one.

And it's something for you
think about a little bit.

It might be obvious to you.

But another way to think about
it is, when you're trying to

take sums of vectors, any other
vector, in order to move

it in that direction, you have
to have a little bit of one

direction and a little bit of
another direction, to get to

any other vector.

And if both of your ingredient
vectors are the same

direction, there's no way to
get to a different one.

Anyway, I'm probably just being
circular in what I'm

explaining.

But that hopefully gives you a
little bit of an intuition of

well, one, you now know what
a singular matrix is.

You know when you can not
find its inverse.

You know that when the
determinant is 0, you won't

find an inverse.

And hopefully-- and this was
the whole point of this

video-- you have an intuition
of why that is.

Because if you're looking at the
vector problem, there's no

way that you can find-- that
either there's no solution to

finding the combination of the
vectors that get you to that

vector, or there are
an infinite number.

And the same thing is
true of finding the

intersection of two lines.

They're either parallel, or
they're the same line, if the

determinant is 0.

Anyway, I will see you
in the next video.