I am a neophyte undergraduate student reviewing multivariate analysis. I am learning from Mathematical Tools for Applied Multivariate Analysis (J.D. Caroll, P.E. Green).
I covered the co-factor expansion method to yield the determinant of a matrix. In my textbook, it is said that for a 3x3 matrix, d(A) is based on the entries of the first row:
- d(A) = a11(a22 * a33 - a32 * a23) - a12(a21 * a33 - a31 * a23) + a13(a21 * a32-a31 * a22)
Which, for the following matrix, yields a value of -441:
2 4 -3
5 7 6
-8 1 9
- d(A) = 2(57) - 4(93) + -3(61) = -441
So far, so good. However, when I moved on to finding the adjoint of the same matrix, I found another method on YouTube where the co-factor entries are applied to the entire matrix, row by row. Meaning that beyond a11, a12, a13 (i.e., first row), the minors and co-factor values are also computed for a21, a22, a23, ..., a33. Which, for the same matrix A above, would yield a co-factor matrix:
cf(A):
57 93 61
39 -6 34
45 27 -6
My first question then is: Why is determinant of A NOT obtained through the computation of all co-factors derived from all row entries of A? Why does the formula for d(A) stops at the first row?
Then, my second question pertains to the transpose of cf(A). In the same video, a signed matrix is used to determine the signs of the transpose of cf(A) (i.e., the adjoint of A).
pos neg pos
neg pos neg
pos neg pos
I am not sure what the transpose rules are in relation to this signed matrix. In my own case study, the co-factor matrix transpose would be, without sign alteration T(cfA):
57 39 45
93 -6 27
61 34 -6
How will the transposed entries change in light of the signed matrix?
Thank you in advance for any kind mind who'll be willing to enlighten me.
Daniel