In previous videos, we've seen how to find the determinant of a
two by two matrix. We've also seen the role that the
determinant plays in solving simultaneous linear equations,
and in finding the inverse of a
matrix. And what we're going to do in this video is we're going
to look at three by three determinant statements of square
matrices with three rows and three columns. Now, if you can
remember the rule for two by two, determinant's was very
straightforward, but perhaps not very intuitively obvious.
With three by three matrices, it gets a little bit more
complicated, and, again not very intuitively obvious and to
workout the determinant of three by three matrices, we need to
first of all introduce some other ideas, called minors and
cofactors, and will start by doing that. And throughout this
video will always use the same matrix A. So this is the matrix.
Say that we're going to use and you can see that it's got three
rows and three columns. We can only find determinants when we
have a square matrix.
The matrix that has the same number of rows as it has
columns. And to find the determinant of this three by
three matrix, we first of all need to understand what's meant
by a minor. Every element within our matrix has its own minor, so
we spell minor like this MINOR, and the minor of an element is
the value of the determinant we get when we cross out the row
and the column containing.
The element we're looking at, so let's look at this element 7.
Element 7 is in the first row and the first column. So if we
cross out that throw in that
column. I'm going to hide them rather than across the Mount.
You see that we are left with a 2 by two matrix and we can find
the determinant of that two by
two matrix. So the minor of the element 7 in Matrix A is
this two by two determinant with entries 3 -- 1 four and
minus 2.
Remember the rule for evaluating a two by two
determinant is to multiply the two elements on the
leading diagonal and then subtract the product of the
two elements that aren't on the leading diagonal. So we
do 3 * -- 2.
And then we take away 4 * -- 1.
So that gives us minus 6 -- -- 4.
She's minus 6 + 4, which gives us minus 2.
So the minor of our elements 7 from our original three by three
Matrix is the value minus 2.
Now we can do the same thing for any element in the matrix and
find its minor. So let's pick this four. The four that's in
the 3rd row and the second column, and we're going to find
the minor of four.
So we go through the same
process. Before is in the bottom row, so we cross that out and
it's in the second column. So we cross that out.
And so we see we're left with a 2 by two matrix.
And so we workout the determinant of that two by
two matrix.
These are two by two Matrix 7, one, 0 -- 1.
So we do 7 * -- 1, which is minus 7 and then we take away
nought times one we're taking away nought. And so we're
getting an answer of minus 7.
So the minor of the element for in this matrix is minus 7.
Now we could, but I'm not going to now. We could workout the
minor of every single element in our matrix A and we get a value
of the minor. So we get 9 values, one for each element of
the matrix A.
Now you can see now what the process is to work those apps.
I want to go now from minus to
cofactors. And to get from minors cofactors,
I need the idea of what we call a place sign.
Every place within the Matrix has a sign associated with it.
We always start with a plus in the top left hand corner and
then we alternate the signs. So as we go across the road, we're
going to go plus minus plus.
As we go down the column, we're going to go plus, minus plus.
You can see that if we fill this in, it will look like this.
This is the matrix of place signs. So for instance, the
element 4 we've looked at before has a place sign of
minus, whereas the seven that we've looked at as a place
sign of plus and we used to play signs to convert minors
into what we call cofactors.
So the cofactor of the element 7 is equal to the place
sign. Times
the minor. That's true for all the
elements. Every element has a cofactor which we find by doing
the product of its place sign with the value of its minor.
So if the element 7, the place sign was plus.
And then at 7 the minor was minus two. So we get plus minus
two, which is just equal to
minus 2. To the cofactor of the element 7 in this matrix A is
minus 2. If you take the element 4, the cofactor.
A full Is equal to its place sign times it's minor, so the
minor of four we saw was minus seven. The place sign is a
minus, so we don't play sign minus times the value of the
minor minus seven, so minus minus seven is plus seven. So
the cofactor of Element 4 in this matrix is 7 when they are
going to calculate the determinant of our matrix A by
using the cofactors.
So if we look at what we've written down here, what I've
done is I've taken our matrix A and I've worked out the 9 minus.
So the minor of seven. We've already seen his minus two and
the minor of four we've seen as minus seven. I've worked out all
the other minors and I've put them into this matrix here that
I've called M. Remember that we have a place sign matrix that
starts with a plus in the top left hand corner and alternate
sign as you go down or across.
And then we calculate the cofactors by taking the minor
and multiplying by its place site. So I've assembled into
the Matrix, see the cofactors of all the elements of our
matrix A and you can see that you get the cofactors from the
minors simply by multiplying by the place site.
Now to workout the determinant
of a. What we have to do is we can pick any row or any column.
So let's say we were to pick the first row and you see that the
row has three elements in it would pick the column that would
have had three elements in it as
well. What we do is we pick our row. We've got three
elements and they've each got their own cofactor.
So we're going to do is going to form the product of element
times its cofactor, so we'll be doing 7 * -- 2 two times, three
under 1 * 9.
I've just explained what the steps are for calculating
determinant, so let's now write
it all down. We've seen that the only things that really matter
are the matrix we started with and its cofactors, so that's all
we've got on on the sheet.
So here's our matrix A.
And here's the matrix of
cofactors. Remember what we need to do is we were picking the
first row. We could pick any row or column, but we've chosen to
pick the first row.
And workout the determinant of
a. Or write Det the determinant, the T brackets a meaning of the
determinant of a.
And sometimes you'll see this written with vertical lines.
To mean the determinant of a.
And what we do, because we've chosen the first row, we write
down the elements of the row 7211, and we multiply each
element by its cofactor. So introduce 7 * -- 2.
We're going to do 2 * 3.
And we're going to do 1 * 9.
So we can get these three products and then we add them
up. So 7 * -- 2 is minus 14. Two times three is 6 of 1
* 9 is 9, so we workout minus 14 + 6 + 9. We get the
answer one. So the determinant of a is equal to 1.
Now you're probably thinking, well, what if I chosen a
different row or a different column? Surely I would have got
a different answer, so let's choose a different row, a
different column, see what happens. Let's choose the second
column. So if we're going to use the second column to workout the
determinant of a, we've got elements 2, three and four, 2,
three, and four.
And we multiply these by their cofactors. 2 gets multiplied by
three, 3 gets multiplied by minus 11, and four gets
multiplied by 7.
We got the three products.
And we add them up.
So 2 * 3 is six 3 * -- 11 is minus
33 and 4 * 7 is
28. Work this out. We get one.
So we get the same value.
And we will always get the same value whichever row or column
that we choose to use.
Doesn't matter, it's completely our choice. So what this means
is that if we're working out the determinant of a matrix, we
actually don't need to workout all nine cofactors.
You see, when we use the first row, we only needed these three
cofactors. When we use the second column, we only needed
those three cofactors.
So before we go ahead and work out all nine, if all we want to
do is workout the determinant of a, we should decide which row or
column we're going to use first, and then only workout the
cofactors that correspond to the row or the column that we've
chosen. And in general it's handy idea. Good idea to keep
your workload down is if you've got zeros in a row or column.
That's a good row or column to choose, because you don't need
to actually workout the cofactor that goes with the zero element,
because it's going to get end up being multiplied by that zero
element. So if we use the first
column. The first value is 7, the next value is zero and the
last value is minus 3.
We then have to multiply these elements by their corresponding
cofactors 7. * -- 2.
Nought Times 8 and 3 * -- 5.
Have them up. We get 7 * -- 2 is minus 14 nought. Times 8 is
nought minus 3 * -- 5 is plus
15. An minus 14 plus not plus 15 is equal to 1. The same answers
before we knew it was going to come out to be the same answers
before. The important thing is that we really didn't need to
know what the code factor of 0 was. So once we've chosen to use
this column, we actually need to find the cofactors of seven and
minus three because the Co factor of nought is going to get
multiplied by normal. So what we've seen then is how to
workout the determinant of a three by three matrix. What we
do is we pick Aurora column and keep our work as low as
possible. We choose the row or column with the most zeros, and
then what we do is we workout the cofactors of every element
in the row or column that we've chosen, but we don't have to
workout the cofactors of zero
elements. Because what we do then is we multiply each
element by its cofactor, and then we add up the products of
element times, code factor, and that's how we workout the
determinant. Now this process works for any size square
matrix, although once you get above 3, the amount of
arithmetic involve gets to be very large.