CHAPTER 04.05: UNARY MATRIX OPERATIONS: Determinant of a matrix using minors Theory
In this segment weíll 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 weíre 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 youíre seeing that the A I† J element is corresponding to a minor called M I J, so this is the minor. And if thatís 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 letís 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. Itís 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, letís suppose Iím 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 thatís how youíre 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 thatís what weíll do next. And thatís the end of this segment.