CHAPTER 04.05: UNARY MATRIX
OPERATIONS: Determinant of a matrix using minors Theory In
this segment we’ll talk about how to find the determinant of a matrix using
minors. So the determinant first can
be only found of a square matrix. So let a be a n by
n matrix. So we’re talking about a square matrix. And then let the minor, the
minor of A I J. So a minor corresponds to each element of the A matrix so the
minor of A I J is denoted by M I J. So you’re seeing that the A I J element is
corresponding to a minor called M I J, so this is the minor. And if that’s
the case then we can find the determinant of A. We can find the determinant
of A by any of these particular formulas. So we can say J is equal to one to n minus one I plus J, A I J M
I J for any I going from one all the way up to n. So let’s break it down what it means. What
it means is that I can find the determinant of the A matrix, the square
matrix by taking minus one raised by I plus J so that is just simply a minus
one or a plus one depending on whether the I plus J is odd or even. Then I take the element of A and then I
multiply it by its minor. But in order to be able to do this summation I need
the value of I, but I can choose any value of I. I
can choose the I equal to one, I can choose the I equal to two, I can choose
the I equal to n, I can choose any value of I going from one to n in order to be able to use this formula. I can also
do the same thing I can find the determinant of A by using any J, means I can
use any column. By
writing this equation as this I is equal to one to n,
minus one raised power I plus J A I J M I J for any J going from one to n. So very similar formula as this one, the only
difference being that this is for any row I, and this can be used by any row
J. The question arises, what is this minor? Well the minor is M I J simply is
a determinant itself. It’s a determinant of n minus one by n minus one matrix
for which the Ith row and Jth
column are removed. So what we are doing here is that if I would have to find
the, let’s suppose I’m looking at A solve for one two then I would have to
find M one two. What that means is I would have to take, I have to get rid of
the first row and the second column. SO M I J simply means that first I find
the n minus one by n minus one matrix in which the Ith
row has been deleted the Jth column has been
deleted. Once this row and this column has been deleted it will result in a n minus one by n minus one matrix and I have to find the
determinant of that in order to be able to use in this formula. So
the common question which is asked is that hey I was just trying to find the
determinant of a five by five matrix and it results in this summation here
which is itself involves a calculation of a four by four matrix determinants.
So what you can see is that if you use this formula again and again it will
result eventually in a determinant of a one by one matrix because you can
write down the determinant of a n minus one by n minus one matrix in terms of
the determinant of n minus two by n minus two matrices and so on and so
forth. And you can eventually get the answer in terms of a termed one by one
matrix. The determinant of a one by one matrix is the matrix itself, the
scalar which is in the first row first column so that’s how you’re going to
use this formula. Now the best way to understand or use or understand this
formula and also use it is through an example and that’s what we’ll do next.
And that’s the end of this segment. |