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## Precalculus

# Invertible matrices and transformations

An invertible matrix is a matrix that has an inverse. In this video, we compare the effect of two transformations, one defined by an invertible matrix and one by a non-invertible matrix. We see that the non-invertible matrix map the entire plane onto a single line. Created by Sal Khan.

## Want to join the conversation?

- Did Sal make a mistake? I don't see how point (1,1) can be transformed to point (4,4) via this transformation at all. Shouldn't it be point (0,0)?(3 votes)
- I believe the (1,1) point represents the resulting vector from the unit vectors (no transformation).

Think of the parallelogram rule for vector addition.

The (4,4) point represents the same resulting vector "under a transformation".

However, this time, the vectors that the resulting parallelogram is built out of are the (2,1) and (2,3) vectors.

I hope this helps!(1 vote)

## Video transcript

- We have two by two matrices here and in other videos we talk
about how a two by two matrix can represent a transformation
of the coordinate plane of the two dimensional plane where this of course is the x-axis and this of course is the y-axis. What we're doing in
this video is visualize these transformations and
get a visual understanding for why it's reasonable
for A to have an inverse or for why matrix A is invertible and why it's not reasonable for B to have an inverse matrix or why matrix B is not invertible. So just as a reminder transformation these transformation matrices essentially tell us what to
do with our unit vectors. For example, we have
the one zero unit vector and the first column of
each of these matrices tell us what the one zero unit vector the vector that goes one
unit in the extraction what it will get transformed to under each of these transformations and then of course you have
the unit vector in the Y direction the zero one unit
vector and the second columns here tell us how we would transform that. So let's first think about
matrix A, matrix A transforms the one zero vector
into the two, one vector so the two, one vector is going
to look something like that and it transforms the zero one vector into the two, three vector
so the two three vector is going to look something like this. So one way to think about
it instead of our grid looking like this grid
that I already had here which is just our
standard coordinate axes. This is defining a new grid
that would look like this and I define this new grid
by looking at our multiples of what the one zero vector
has been transformed into and we also look at multiples
of what the zero one vector has been transformed into. So for example, if I were to
take this point right here before it is transformed
it has one of each of these vectors, well under this
first transformation it would be one of each of these vectors. So it's going to be one two one vector plus one two three vector
so this point will be mapped to this point and by that
same logic this point right over here which is one
unit more in the extraction well now it's going to have one unit more in the direction that the X unit vector has been transformed into
this point right over here by the same logic will be one
unit more in the direction that the Y unit vector
has been transformed into so it will be there. And that this point by the
same logic will be transformed into that point and so this
region that I'm showing you in white will get
transformed to this region. Now there's some obvious
things going on here. We have a two dimensional
area that has been transformed to another two dimensional
area and in fact, it looks like it has been scaled up in
other videos we have talked about that this scale factor
is going to be the absolute value of the determinant
of A and it's clear that frankly, not only is this non-zero but it's going to be greater
than what it looks like we are scaling up our area
but the very fact that this does not equal zero tells
us that we were scaling from a two dimensional area to
another two dimensional area. And so it's completely
reasonable to be able to go back you will, for sure be able
to find a transformation that takes you from this
region to that region. So that makes us feel good
that A inverse is reasonable. Now, as a point of contrast
let's think about matrix B right over here. Matrix B transforms the
one zero unit vector into the two, one unit vector so it transforms it into this unit vector actually very similarly to how A did it but then let's see what it
does to the zero one vector transforms this vector
into the four two vector four two is this notice
the four two vector is just two of the two one vectors is going in the same direction it just has a different
size or different length or a different magnitude. So in this situation is taking things in two dimensional space
and it's turning them into combinations of things
that sit on this same direction. So everything in two dimensions
is going to be mapped to something along this line here. So this region, if I were
to apply transformation B it's going to be a region
of this line over here anything, if I were to
map it using matrix B is going to get mapped onto this line so you're going to something with area and then you're mapping it
to something that has no area so the scaling factor here must be zero. So we know since the absolute value of the determinant of B is
zero, we could also fairly say that the determinant of B is equal to zero and the how are you going
to have an inverse matrix? How are you going to have
something that can scale from zero area to something that has area? So we know that B inverse does not exist there's nothing that can transform us back and so there's a couple of things here this reinforces this idea
that if your determinant of a transformation matrix is zero you're not going to have an
inverse, it's not invertible the other thing to recognize
is seeing the patterns in the matrix itself here we
saw that the second column is just a multiple of the
first column it's twice the first column two times two
is four one times two is two you can also view it the other way round the first row is a
multiple of the second row and you can play around
with the math if you wanted to generalize it and
see if that's the case the determinant will always be zero and that's because if you
view them as transformations it's going to map to line
and you're gonna lose all your area, if you view
it as a representation for at least the left-hand
side of a system of equations you can think about it as
lines that have the same slope but we talk about that in other videos.