I am currently studying linear algebra. So far I have studied method for aquiring the inverse of a matrix A. Now I would like to evaluate the inverse of a $4\times 4$ matrix using the following formula:
$$
\mathbf{A}^{-1} = \frac{ \mathrm{adj}(\mathbf{A}) }{ \det(\mathbf{A}) } % A^(-1)=1/det(A) adj(A)
$$
My question is in order to get the adjoint matrix I will need the cofactor matrix. In a $4\times 4$ matrix this means I could do it with row expansion in 2 dimensions. For instance, I take the entry A[a11], I cross out first row and first column. Next I choose B[a11] in the submatrix to get my minor. Now, do I have to multiply the minor of the submatrix with the original entry A[a11]? Does it matter what minor I choose in the submatrix in further calculation? For example, using A[a12] to get the minor of the main matrix, can I use any row or column in the submatrix?
I appreciate any help in this manner. Thank you.
-Daniel
Best Answer
If I understand your question correctly, when calculating the adjoint matrix, you do not need to multiply the determinant of the submatrix by the entry you're finding the cofactor for.
One reason this can be confusing is that you do multiply the submatrix determinants by the entries to find the determinant of the matrix.