Algebra: Define the cofactor C2,3 for a 3x3 determinate?

Question

any help will be appreciated

Answer 1

If you strike out the second row and third column of your 3 x 3 matrix of which you are trying to find the determinate, a 2 x 2 matrix will be defined by the numbers in the remaining rows and columns. The cofactor is the determinant of this smaller matrix, multiplied by either -1 or 1, depending on the sum of the row and column numbers. If this sum is even, you multiply by 1; if odd, by -1. In this example, the sum is odd so you multiply by -1.

Answer 2

Take row 2 column 3. 2 + 3 =5 is odd, so the
sign of the minor of a23 is -.
Now scratch out row 2 and column 3 and
take the negative of the remaining 2*2 determinant.
The cofactor of any element is its signed minor.
Let's do a simple example
Suppose we have the determinant
1 2 3
4 5 6
7 8 9
and we want the cofactor of a23 = 6
It is
- | 1 2| = 6
..| 7 8|

Hope that helps!

Answer 3

Cofactor of an element of a 3x3 matrix is given by [(-1)^(i+j)]M, where i and j are the row and column of the element respectively, and M is the minor of that element.

Minor of an element (which is present in a particular row and particular column) is defined as the determinant of the matrix obtained by deleting that particular row and column.

Thus C23 would mean, cofactor of the element present in the second row and third column.
Therefore, C23 = (-1)^(2+3) M23
= (-1)^5 M23
= - M23
where M23 is the minor of the element present in the second row and third column. i.e the determinant of the matrix obtained by deleting the second row and the third column.