Every square matrix has
associated with it a special

quantity called its determinant.

And determinants turn out to be
very useful when it comes to

doing more advanced things with
matrices like finding inverse

matrices and solving
simultaneous equations and later

videos will cover these topics
were going in this video. Just

look at finding the determinant
of a two by two square matrix.

So let's look at an example.

Suppose we have a matrix A.

Which is got entries
in it elements in it

435 and minus one.

So this is a two by two matrix.
It's a square matrix.

Now the determinant is a single
value. It's a value that's

generated by combining the
numbers within the matrix in a

special way for two by two
matrices. The way isn't

particularly hard, but equally
it's not particularly obvious

that you would do it this way,
so we need to just see an

example, so we want to find the
determinant of A and we normally

denote that by the letters Det
being the first 3 letters of

determinant. So we right.

Debt of a.

A moment working out
determinants instead of using

round brackets around the
matrix, we use vertical lines.

But the numbers within the
vertical lines are exactly

the same.

And now we have to combine
those numbers.

The first thing we do is we look
at the numbers on the leading

diagonal. That's the four and
the minus one. And what we do is

we find their product, we

multiply them together. So 4 *
-- 1 Now having dealt with the

numbers on the leading diagonal,
we've got 2 numbers left. The

numbers that aren't on the
leading diagonal, and so we

multiply those two numbers
together. So that's 3 * 5.

So we've got the product of the
numbers on the leading

diagonal, and we've got the
product. The numbers that are

not on the diagonal, and then
we just take this second

product away from the first
product.

And then we work it out 4 * -- 1
is minus four. 3 * 5 is 15, so

we take away 15 -- 4 takeaway.
15 is minus 19.

And so that's how we do every
single determinant for a two by

two matrix. We simply find the
product of the numbers on the

leading diagonal and take away
from that the product of the

other two numbers. So let's do

another example. Here's another
two by two matrix B with

elements 6, two.

Three and five.

And to workout the
determinant of B.

We find the product of the
numbers on the leading diagonal

is 6 and five. Then we take away
the product of the two numbers.

Those two numbers are two and

three. So we're taking
away 2 * 3.

So 6 * 5 is 32 *
3 is 630 takeaway 6, and

we're getting 24.

The one more example here. So we
got the matrix D.

Which has got entries 64.

Three and two. Now when we
workout, the determinant of D.

New the product, the numbers on
the leading diagonal is 6 * 2.

Then we take away the product of
the two numbers, which is 3 * 4.

So 6 * 2 is 12. Taking away 3
* 4, she's also 12, and so we

get the answer 0.

So we see that every time we
workout the determinant, what we

get is a single number.

Determinant of a is minus 19 the
determinant of B is 24

determinant of D is 0.

Now when a matrix has a zero
determinant, it has a especially

given name to that property. We
say that the matrix is singular.

So any matrix which is singular
is a square matrix whose

determinant is 0.

These other matrices, where
the determinant wasn't zero

are called nonsingular.

So it doesn't matter what the
determinant is, just so long as

it isn't zero. That makes the

matrix nonsingular. Finally,
we'll just look at a general two

by two matrix. So here we have a
matrix A and the entries the

elements in this matrix RABC&D.

If you want to workout
the determinant of a.

We just follow the process we've
already seen. We take the values

that are on the leading
diagonal. We multiply them

together to get a D. We take the
other two values and multiply

them together. It should be in C
and we subtract this second

product away from the first
product, and so this is the

general formula for any two by
two matrix. The elements are

ABCD. The determinant is worked
out to be a * D minus.

B * C.

So that's all there is to know
about 2, but the determinants of

two by two matrices.