https:/.../video3-additionetcf61mb.mp4

In this video we look at the
subjective addition and

subtraction of matrices, and we
also look at scalar

multiplication of matrices. To
do that, we're going to need

some matrices, so here are some
matrices that I've already

prepared, and you'll see we've
got four matrices here, and

they've all got different sizes.
So the first thing we need to do

is just remind ourselves as to
how we look at the size of a

matrix, so we count up the
number of rows and the number of

columns. So this matrix A.

We say is a two by two matrix
because it's got two rows and

two columns. Matrix B's got
three rows and two columns, so

that's a three by two matrix.

And we can clearly see that
matrix C is a two by three

matrix, two rows and three
columns, and matrix D is a three

by two matrix with three rows
and two columns.

Now, when it comes to adding and
subtracting matrices, we can

only do it when the two matrices
have the same size. That is,

when they got this both got the
same number of rows and the same

number of columns and two
matrices that have the same size

are said to be compatible, and
when they're compatible we can

add them and subtract them.

So if we return to our four

matrices. We see that of these
four matrices, the only two that

are compatible or matrix B and
matrix D. They have the same

size 3 rows and two columns. So
what that means is that we can

find B + D and we can find
B -- D and we can find D

-- B. So we can add and subtract
matrices, B&D because they have

the same size.

Because they're compatible, we
can't add A&B because they have

different sizes. We can't add
C&A because they have different

sizes. Might be worth noting
that where we define the matrix

C transpose. C transpose. That's
where the rows become columns

would have. Two columns, because
each row would turn into a

column, it would have three
rows, so C transpose would be a

three by two matrix. So C
transpose is also compatible

with B&D. So we can add C
transpose to be or today, but we

can't add C to be or today.

Once we found two matrices that
are compatible that these two

matrices that have got the same
size, then we need to know how

to actually add them up. So we
look at our matrices B&D and see

how we go through this process.
So here's our Matrix B and

here's our Matrix D and I've
written them with a plus sign

between them and underneath I've
written them out again with a

minus sign. So this is B + D and
this is B -- D. So how do we do

the addition? Well, it's quite

straightforward. All we do
is we were adding we add the

elements that are in the
same position. We call that

the corresponding position.

So because the five is in the
first row on the 1st column and

the two is in the first row and
the first column, they get added

together. So we do 5 + 2 which
is 7 and that gives us the entry

in the first row and the first
column of our answer.

We do the same with all the
elements, so the minus one is in

the 2nd row and the first
column. So we add that to the

zero in the 2nd row and the
first column. So we do minus 1 +

0, which gives us minus one and
we can continue to do that for

all six elements of the matrix.
So 1 + 4 because the one and

four are in corresponding
positions gives us 5 -- 2 + --

2. Gives us minus four and that
goes up here because it's in the

first row and the second column
first row on the second column

for three and the one get added
to give us four and the nought

and the minus one get added to
give us minus one.

And so that's how we do matrix
addition. So just to recap, we

have to have two matrices that
have the same size and then when

we have two matrices at the same
size we add them by adding the

elements that are in
corresponding positions. And so

the answer we get is the same
size as the two matrices that

we've added together.

Now the principles of
subtraction are exactly the

same. We deal with elements that
are in the corresponding

positions, but obviously this
time we subtract rather than add

them. So we do 5 -- 2 to get
three, we do minus 1 -- 0 to get

minus one. We do 1 -- 4 to
get minus three. That's done the

elements in the first column
with the elements in the second

column minus 2 -- -- 2 becomes
minus 2 + 2, which is 0.

3 -- 1 gives us 2
and 0 -- -- 1 is 0

+ 1 which is 1.

And there's our answer. So
when we do B -- D, This is the

answer. Again, a matrix of the
same size as B&D, so that

illustrates how we do matrix
addition and subtraction. We

have to have two matrices
which have the same size in

order to be compatible. And
then what we do is we add or

subtract the elements that are
in the same positions. We call

corresponding elements.

Now Matrix obviously has the
same size itself, so we can

always add a matrix to itself,
and we're going to do that now

with the Matrix A. So we're
going to add matrix A to

itself, so into a plus a. So
here's Matrix A and what adding

matrix a onto it. And because
it's the same matrix, clearly

it's not the same size that
both 2 by two matrices, so we

go through the standard
procedure.

When we add elements that are
in corresponding positions, so

the four gets added to the
four, which gives us 8, the

three gets added to the three
to give us 6 not getting to

nought, which gives us nought
and minus one gets added to

minus one. To give this minus
2.

So matrix a + A is this matrix
here with entries 860 and minus

two and we used to writing A
plus a in a shorthand form as a

+ A = 2 A.

One lot of a there's another lot
of a gives us two lots of a.

So this matrix that we found
here, we can refer to as 2A.

And if we look at the entries in
this matrix and compare them

with the entries of a, we see
that each of the entries is just

twice the entries of a 2 * 4 is
eight 2 * 3 or 6 two times

naughties nought 2 * -- 1 is
minus two, and so this process

illustrates how we do we call
scalar multiplication. We take a

matrix and we multiply it by a
number. All that happens is that

every element inside the matrix.

Gets multiplied by the number,
so in this case the number was

two and we'll do some examples
now, but we use a different

number. So we've seen how we can
do scalar multiplication by

simply multiplying every element
inside our matrix by the number.

The scalar that we're trying to
multiply by. So we'll do a

couple more examples now, so
we're going to workout is going

to five times the matrix B and
I'm going to do 1/2 times the

matrix D. So all I've done is
I've written down what matrix B

is. I'm going to do five times
this matrix. So remember the

rule for scalar multiplication
is the scalar, the number that

we're trying to multiply by
multiplies every entry inside

the matrix. So we get 5
* 5 is 20 five 5 * --

1 is minus five. 5 * 1 is

5. 5 * -- 2 is minus
ten. 5 * 3 is 15 and 5

* 0 is 0.

So this is our answer. This is
the matrix 5B or scalar five

times matrix B. Notice that the
matrix we get to that answer has

the same size, the same order as
the matrix we started from, and

that's fairly obvious. That must
be the case because all we do is

we multiply every element inside
the matrix by the scalar

outside. So we are not creating
any new entries in the Matrix

and we're not losing any. So the
matrix that we get must have the

same size. So when we
started with.

Here's another example. We don't
have to just multiply by whole

numbers or two previous
examples. We did 2 * A and 5 *

B, but we can multiply by any
number, and in this case I'm

choosing to multiply the
fraction or half. So I'm going

to do half times matrix D.

So here it is written out.
Here's matrix D. We do 1/2 times

that number. All we have to do
is do 1/2 times every element

inside the matrix.

So we do 1/2 * 2, which gives us
one 1/2 * 0, which gives us

nought 1/2. Times 4, which gives
us two. That's not the first

column and 1/2 * -- 2, giving us
minus 1/2 * 1, giving us a half

and 1/2 * -- 1, giving us minus

1/2. And so here's our product
matrix and not surprisingly, it

ended up with some fractions.
Then, because we were

multiplying by a fraction to
start off with.

That concludes the video on
addition and subtraction of

matrices and on scalar
multiplication.