CHAPTER 04.08: UNARY MATRIX OPERATIONS: Determinant of a matrix using cofactors Example
Letís take an example of how to find a determinant of a matrix using the cofactor method. So letís suppose somebody says hey go ahead and find the determinant of this matrix: 25, 5, 1 64, 8, 1, 144, 12, 1. So what weíre going to do is we got to find the determinant of this matrix so weíll 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 letís suppose I know n is 3 because thereís three of those in three columns, and letís 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, Iím 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.
Letís 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 itís minus one raised power I plus j times the whatever is the minor corresponding to the first row first column. So thatíll be minus one raised power one plus one is just positive so thatíll 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 youíre left with 8, 1, 12, and 1. And that is nothing but eight times one minus twelve times one and you get minus four. Letís 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, Iím left with 64 one, 144 and one. Sixty-four and one and one, thatís what Iím left over with. So that gives me minus, and then Iíll have 64 times 1 minus 144 times one and that gives me the value of 80. So thatís what turns out to be C one two. Letís find what C one three is. Itíll 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 thatíll be the determinant of the matrix which is left over one side moved first row third column.†
So Iím 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? Itís 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 thatís the end of this segment.† ††