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.