CHAPTER 04.07: UNARY MATRIX
OPERATIONS: Determinant of a matrix using minors Example In
this segment we’ll look at an example how to find determinant of a square
matrix by using minors. So let’s suppose somebody says hey find determinant
of this matrix here: 25, 5, 1, 64, 8, 1, 144, 12, 1. So the determinant of a
matrix is given as simply as if I’m going to find it out in terms of my any
of the rows, will be minus one times I plus J, raised to I plus j, A I J
times M I J, for any I is equal to 1, 2, and 3. So this is a three by three
matrix, I can choose any row so I can choose I equal to one two or three. And
then do this summation from J is equal to one to three. Well A I J is simply
the coefficient of the entry I J of entry of the A matrix. And M I J is the corresponding minor to
that particular entry. So let’s suppose I take the first row as my I. So
let’s suppose I choose I equal to one. So what does that mean? What do I get
for the determinant of A? I get determinant of A
equal to summation J is equal to one to three minus one raised power one plus
J, A 1 J times M 1 J. So that’s the formula which I’ll have to use in order
to be able to find out what the determinant of A is. So let’s go and see
what, So J would be the values of one to three so basically what that means
is that I would have to find M one one, M one two
and M one three. So let’s go and see
what M one one is. So M one one
minor would be simply equal to that I take out the first row and first column
and find the determinant of the rest of it. So M one one
would be take out the first row and the first column and whatever is left over find the determinant of that,
so that’s the determinant of what is left is eight one twelve one. That will be this multiplied with this
minus this multiplied with this. So its eight times one minus one times
twelve and that gives you minus four. So that’s M one one. Let’s
go ahead and find out the other minors as well. So M one two will be you take
out the first row and the second column out and that’ll be the determinant of
the matrix if you take the first row and second column out you’re left with
sixty-four, one and one forty four and one. So that gives you sixty four
times one minus one times one forty four and that gives you minus eighty. So
that’s M one two, similarly lets go and find out what M one three. So M one three will be you take out the
first row and you take out a third column and whatever is left over you find
out what the determinant is of that. So if you take the first row out of the
third column out you get sixty four eight, one forty four twelve. And that’s
this times this minus this times this. Sixty four times twelve minus eight
times one forty four equals minus three eighty four. So
we have found out all the minors which we need now, so as we’re talking about
the determinant of A will be equal to summation J is equal to one to three
minus one raised power one plus J A I J times M 1 J. SO if we expand this
we’re going to get for the value of J equal to one, one plus one minus one
raised to the power of two will be equal to a positive number. So I’ll get A
one one here and M one one
here plus for J equal to two I’ll get one plus two is three for minus one
raised to the power of three would be a negative number so I’ll get A one two
M one two. It would be minus here, it would be minus here because it is minus
one raised power one plus two is minus one raised to the power of three and
that’s a negative number so it would be minus. Then the third one, third term
which is J equal to three would be one plus three. One plus three is four so
minus one to the power of four is plus one. So it’ll be plus A one three M
one three. So that is what I will get
as my determinant. So Ace of one one is nothing but
25. And the first minor M one one which we
calculated was minus four. Then A one
two is five. And M one two which I calculated is minus eighty. Plus A one
three is one and the M one three which I calculated is minus 384. If I do
this multiplication and the addition and subtraction operations I get minus
100 here, minus plus 400 here and minus 384 and that number turns out to be equal
to minus 84. So
that’s how we find the determinant of the matrix values and minors. You could
very well use any row or any column to find the determinant by using minors.
Here we’ll just use the first row. You
can repeat the process by using the second row or third row or by first
column second column and third column and that’s up to you. And still the
number that you get is minus 84. And that’s the end of this segment. |