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# Multiplying matrices by scalars

Video transcript

Now that we know
what a matrix is, let's see if we
can start to define some operations on matrices. So let's say I have the 2 by 3
matrix, so two rows and three columns, and the entries are
7, 5, negative 10, 3, 8, and 0. And I want to define what
happens when I multiply 3 times this whole thing. So first of all, let's get
a little terminology out of the way. The number three, in
just the everyday world, if you weren't dealing
with matrices or vectors, and if you don't know
what vectors are, don't worry about
them just now, you would just call that a number. You would call
this a real number. It's just a regular
number sitting out there. But now in the
world where we have these new structured
things, these matrices, these arrays of
numbers, we will refer to these just plain
old real numbers that aren't part of some
type of an array here, we call these scalars. So essentially what
we're defining here, we don't know-- I haven't
said what this is actually going to turn out to be,
but whatever this turns out to be will be a product
of scalar multiplication, where we're multiplying
a scalar times a matrix. And so how would
you define this? What do you think
this should be? 3 times this stuff
right over here. Well, the world could have
defined scalar multiplication however it saw fit, but
one way that we find, perhaps, the most obvious
and the most useful, is to multiply this
scalar quantity times each of the entries. So this is going to
be equal to 3 times 7 in the top left, 3 times
5, 3 times negative 10, 3 times 3, 3 times
8, and 3 times 0, which will give us--
it didn't change the dimensions of the matrix. It didn't change,
I guess you could say, the structure
of the matrix, it just multiplied each
of the entries times 3. So the top left
entry is now going to be 21, the entry
in the middle row, top column is going to be
15, negative 30, 9, 24 and 0. So when you multiply a
matrix times a scalar, you just multiply each of
those entries times that scalar quantity.