What is the correct method for calculating the co factor of a 3 x 3 matrix?

[Rob K]

This is insane, I am trying to revise the inverse of matrices and this one element is being really stubborn, please help.

Here is the matrix
3 -1 7

2 0 1

5 -2 6

I have transposed it
3 2 5

-1 0 -2

7 1 6

Now as for replacing the element of the co factors, I am stuck on the second one on the first row.

I have calculated the co factor to be as follows:

(-1 * 6) - (-2 * 7) = 8
fine
but now you times it by the place sign, which is a minus is it not?

So this gives me -8

the book disagrees
And to my horror so does Matlab.

Where am I going wrong?

An help would be appreciated, as I simply can't see where I am going wrong.

Kind regards

Rob

 

[DivisionByZro]

Let me clarify first, are you trying to find the co-factor expansion along a specific column or row? If so, which one is it? Recall that the co-factor expansion formula requires any position where the sum of the row index and column index being odd to be of negative sign (Ex: Suppose your matrix is A, A_1,2 has a negative because 1+2=3 which is odd).
Are you taking the determinant to check whether the matrix is indeed invertible?

 

[HallsofIvy]

Start at the upper left of the matrix, count, starting with "0" as the first element, across the row to the correct column, then count down to the element you want. The sign is $(-1)^n$ where "n" is the final count.

Alternatively, and perhaps simpler, is to say "plus" at the upper left and go across and down alternating "minus" and "plus" at each element.