How is the determinant of a determinant equal to the determinant of the matrix to the power of its number of rows?
How is the following true?:
det((det(A))I) = (det(A))^n(det(I))=(det(A))^n
I understand that det((det(A))I) = det(det(A))det(I) and that the determinant of the Identity matrix is 1, but don't understand the relation that det(det(A))=(det(A))^n.
- JOHNLv 76 months agoFavorite Answer
det(A)I is a diagonal matrix with elements in the leading diagonal each of value det(A). And the determinant of a diagonal matrix is the product of the diagonal elements. This is how you get (detA))^n. Also note that det (I) = 1. The whole line det((det(A))I) = (det(A))^n(det(I))=(det(A))^n follows from this.