A is a non singular square matrix of order 2 such that |A + |A|adjA| = 0, where adjA represents adjoint of matrix A, and |A| represents det(A) (determinant of matrix A)
Evaluate |A – |A|adjA|.
I believe this can be solved using Cayley Hamilton Theorem, but I'm not sure exactly how one would proceed to find the value of the required determinant. The motivation behind my thought was appearance of |A - kI| = 0 in the determinant to be evaluated, where k is a constant.
#1 Of course, |A|adjA can be reduced to |A|² A-1, leaving us with |A + |A|² A-1|. I don't know how to take it from here. A detailed solution using Cayley Hamilton theorem would be appreciated.
#2 I guess that for a square matrix of order n, the required determinant would be equal to n². Could someone please help prove or disprove the result, by generalising for a nxn matrix?
Thanks a lot!