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## Transpose of a matrix

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# Transposes of sums and inverses

## Video transcript

Let's see if we can prove to
ourselves some more reasonably interesting transpose
properties. So let's define some matrix C,
that's equal to the sum of two other matrices, A and B. And so any entry in C, I can
denote with a lowercase cij. So if I want the ith row in jth
column it would be cij, and so each of its entries are
going to be the sum of the corresponding columns that
are matrices A and B. So our ij entry in C is going to
be equal to the ij entry in A, plus the ij entry in B. That's our definition
of matrix addition. You just get the corresponding
entry in the same row and column, add them up, and you get
your entry in the same row and column, and your new
matrix is the sum of the other two. Now, let's think a little bit
about the transposes of these guys right here. So, if A looks like this. I won't draw all
of the entries. It takes forever. But each of its entries are
ij, just like that. Let's say that A transpose
looks like this. Each of its entries, we would
call it, that's if you've got that same entry, we're going
to call it a-prime ij. And these things aren't probably
going to be the same. There's some chance they are,
but they're probably not going to be the same. But that its ijth entry. In the ith row, jth column. In A transpose. Now, the fact that this is the
transpose of that means that everything that's in some row
and column here is going to be in that column and row over
here, that the rows and columns get switched. So we know that we could write
that a-prime ij, we're going to have the same entry
that was in aji. Maybe aji is over here.
aji is over here. So, this thing over here, which
is in the same position as this one, is going to be
equal to this guy over here if you switched the rows
and columns. I think you can accept that. And you can make the same
argument for B. Let me actually draw it out. So if I make B transpose. The entry in the ith row and
jth column, I'll call it b-prime ij. Just like that. Just like I did for A. So we could say that b-prime ij
is equal to, you take the matrix B, what's going to be
the entry that's in the jth row and ith column. These are, you could
almost say, the definition of the transpose. If I'm in the third row and
second column now, it's going to be what was in the second
row and third column. Fair enough. So we already have what
cij is equal to. What's the transpose of cij
going to be equal to? Let me write that down. So C transpose, let me
write it over here. Write C transpose is equal to. I'll use the same notation. The prime means that
we're taking entries in the transpose. So C transpose is just going to
be a bunch of entries, ij. And I'll put a little prime
there showing that that's entries in the matrix
of the transpose, and not in C itself. And we know that c-prime
ij is equal to cji. Nothing new at all. We've just expressed kind
of the definition of the transpose for these
three matrices. Now what is cji equal to? So let's focus on this
a little bit. What is cji equal to? We know that cij is equal to a
sub ij plus b sub ij, so if you swap them around, this is
going to be equal to, you just swap the j's and the i's. a sub ji plus b sub ji. I just used this information
here-- you could almost view it as this assumption or
this definition-- to go from this to this. If I had an x and a y here, I'd
have an x and a y here, and a x and a y here. I have a j and an i here, so I
have a j and an i there, and a j and an i right there. Now what are these? What are these equal to? This is equal to. This guy right here is equal
to-- we do it in the green-- the same entry for the
transpose of a at ij. And this is equal to the
same entry for the transpose of b at ij. Now, what is this telling us? It's telling us that the
transpose of C, which is the same thing is A plus B, so it's
saying that A plus B, A plus B transpose is the same
thing as C transpose. Let me write that. C transpose is the same thing
as A plus B transpose. So these are the entries in A
plus B transpose right here. And what is this over here? What are these? These are the entries
right there. We do the equal sign
over here. What are these? These are the entries in A
transpose plus B transpose. Right? These are the entries
in A transpose. These are the entries
in B transpose. If you take the sum of the two,
you're just adding up the corresponding entries. So that's straightforward to
show that if you take the sum of two matrices and then
transpose it, it's equivalent to transposing them first, and
then taking their sum. Which is a reasonably
neat outcome. Let's do one more and I think
we'll finish up all of our major transpose properties. Let's say that A inverse-- this
is going to be a slightly different take on things. We're still going to
take the transpose. So if we know that A inverse
is the inverse of A, that means that A times A inverse
is equal to the identity matrix, assuming that these
are n-by-n matrices. So it's the n-dimensional
identity matrix. And that A inverse times A is
also going to be equal to the identity matrix. Now, let's take the
transpose of both sides of this equation. I'll do them both
simultaneously. So if you take the transpose of
both sides of the equation, you get A times A inverse
transpose is equal to the identity matrix transpose. And what's the transpose
of the identity matrix? Let's draw it out. The identity matrix
looks like this. You have just ones all the
way down the diagonal and everything else is 0. Right, and you could view this
as i 1, 1 i 2, 2 all the way down to i n, n. Everything else is 0. So when you take the transpose,
you're just swapping out the
zeroes, right? These guys don't change. The diagonal does not change
when you take the transpose. So the transpose of the identity
matrix is equal to the identity matrix. And so we can apply that
same thing here. Let's take the transpose
for this statement. So we know that A inverse times
A transpose is equal to the identity matrix transpose,
which is equal to the identity matrix. And then we know what happens
when you take the transpose of a product. It's equal to the
product of the transposes in reverse order. So this thing right here we
can rewrite as A inverse transpose times A transpose,
which is going to be equal to the identity matrix. You could do the same
thing over here. This thing is going to be equal
to A transpose times A inverse transpose, which is also
going to be equal to the identity matrix. Now, this is an interesting
statement. The fact that, if I have this
guy right here, times the transpose of A is equal to the
identity matrix, and the transpose of A times that same
guy is equal to identity matrix, implies that A inverse
transpose is the inverse of A transpose. Or another way of writing that
is if I take A transpose, and if I take its inverse, that is
going to be equal to this guy. It's going to be equal to
A inverse transpose. So, another neat outcome dealing
with transposes. If you take the inverse of the
transpose, it's the same thing as the transpose
of the inverse.