CHAPTER 04.07: UNARY MATRIX OPERATIONS: Determinant of a matrix using minors Example
In this segment weíll look at an example how to find determinant of a square matrix by using minors. So letís 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 Iím 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 letís suppose I take the first row as my I.
So letís 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 thatís the formula which Iíll have to use in order to be able to find out what the determinant of A is. So letís 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 letís 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 thatís 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 thatís M one one.†
Letís 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 thatíll be the determinant of the matrix if you take the first row and second column out youíre 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 thatís 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 thatís 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 weíre 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 weíre 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 Iíll get A one one here and M one one here plus for J equal to two Iíll get one plus two is three for minus one raised to the power of three would be a negative number so Iíll 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 thatís 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 itíll 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 thatís 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 weíll 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 thatís up to you. And still the number that you get is minus 84. And thatís the end of this segment.