CHAPTER 04.06: UNARY MATRIX OPERATIONS: Determinant of a matrix using cofactors Theory
In this segment we’ll talk about how we’ll find a determinate matrix using cofactors. So let be n by n matrix. So because we know that we can only find determinants of square matrices so it has to be a n by n matrix. Then if C I J is the cofactor corresponding to the A I J element, then the determinant of the matrix A is can be found by summation I is equal to 1 to n A I J C I J for any J equal to one all the way up to n. Or you can also use a different formula determinant of A in terms of the j in terms of columns, so j is the one to n a I j c I j for any I equal to one all the way up to n.
So what this means is that we have to be able to find out what the cofactor is corresponding to each of the elements of the IJ, not each element but to correspond some elements of the A matrix as will be evident from this formula here. So what that means is that we can find the determinant of A by doing this summation first by calculating this product for each value of AIJ times C I J for I going from one to n. But we can use any value of J, we can use any value of J from one to n. Or what we can do is we can calculate the summation of J equal to one to n but then we can use any value of I going from one to n. So the product of those two numbers to this summation will give us the determinant of the matrix.
So the big question is the hey what is CIJ? C I J is the cofactor and that’s nothing but minus 1 raised to I plus j times M I j. M I J is the minus. SO in another segment we’ll talk about what we mean by the minor of a matrix of a square matrix. So we should be able to take that segment and look at what we will be the minor corresponding to the A I J element. And the only difference between the minor and the cofactor is this sign. Minus one raised to I plus j. If I plus j is even then the cofactor is the same as the minor correspondent to the A I J element. If the I plus j is a odd number then the only difference is that the cofactor is the negative of the minor correspondent of the A I J element. So if you look at that previous segment you will know what how M I J is defined. Then you can automatically know how the cofactor correspondent IJ element is defined. And that’s the end of this segment.