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.