CHAPTER 04.08: UNARY MATRIX
OPERATIONS: Determinant of a matrix using cofactors Example Let’s
take an example of how to find a determinant of a matrix using the cofactor
method. So let’s suppose somebody says hey go ahead and find the determinant
of this matrix: 25, 5, 1 64, 8, 1, 144, 12, 1. So what we’re going to do is
we got to find the determinant of this matrix so we’ll say the determinant of
A will be equal to summation J is equal to one two n A I j C I j for I equal
to 1, 2 all the way up to n. So I can chose any value I, in order to be able
to find out what the determinant is. So let’s suppose I know n is 3 because
there’s three of those in three columns, and let’s suppose I is equal to 1.
So In that case the determinant of A will turn out to be summation j is equal
to 1, 2, 3 a 1 j C 1j. So j is going from 1 to 3, I’m writing down the
determinant in terms of the cofactors corresponding to the elements in the
first row. So for expand this I get A one one C one
one plus A one two C one two plus A one three and C
one three. So what I will now do is to I already know what A one one, A one two and A one three are which are twenty five,
five and one. But I need to find the corresponding cofactors to those
elements. And once I have those I should be able to find out what the
determinant of this matrix. Let’s
find out the cofactors for each one of those. So C one one
is minus one raised power of one plus one M one one.
So it’s minus one raised power I plus j times the whatever
is the minor corresponding to the first row first column. So that’ll be minus
one raised power one plus one is just positive so that’ll be m11. And what is
m11, m11 is the determinant of the matrix which is left over once we take
care of the first row and first column. You get rid of the first row and the
first column you’re left with 8, 1, 12, and 1. And that is nothing but eight
times one minus twelve times one and you get minus four. Let’s find C one
two, the cofactor corresponding to the A one two element will be minus one
raised power I plus j which will be one plus two times the minor correspondent
to one two which is M one two so this is minus M one two. So
the minus the determinant of hey what I want to do is I want to get rid of
the first row second column. When we get rid of the first row second column
here, I’m left with 64 one, 144 and one. Sixty-four and one and one, that’s
what I’m left over with. So that gives me minus, and then I’ll have 64 times
1 minus 144 times one and that gives me the value of 80. So that’s what turns
out to be C one two. Let’s find what C one three is. It’ll be minus one based
power one plus three I plus j times M one three. And that gives me plus M one
three because minus one is before we positive one. So its M one three and
that’ll be the determinant of the matrix which is left over one side moved
first row third column. So
I’m left with 64, 8, 144 and 12. And what is the determinant of this matrix
will be 64 times twelve minus 8 times 144 and that number here turns out to
be minus 384. So we are calculating the cofactors which we need in order to
find the determinant of A, so the determinant of A we said is A one one times C one one plus A one
two times C one two plus A one three times C one three. So what is A one one? A one one is twenty five.
C one one is what? Minus four. What is A one two?
It’s five. What is C one two? Which we just found out was 80. What is A one
three? A one three is one. And what is C one three? C one three is minus 384.
And we get minus 84 right there as the determinant of the matrix. Now keep in
mind that we can find out the determinant of the A matrix by using I equal to
one, two or three. We can also equal J is equal to one, two or three. So all
of those different methods will come up with the same determinant. So we can
expand your determinants in terms of any row or column. And that’s the end of
this segment. |