Given , an matrix, with the following characteristic polynomial
show that is invertible if only if
what I have is f(t) can be written as where is the identity matrix, so I would have:
now from the definition of invertible there must exist a matrix which is matrix such that
but how do I find such a matrix, and what is the connections with , beyond the fact that it can't be 0?
I would have pointed out that every matrix is either diagonalizable, with its eigenvalues on the main diagonal, or has a Jordan form, again with its eigenvaues on the main diagonal, both of which have determinant equal to the products of the eigenvalues. Since the determinant is the product of the eigenvalues, a matrix is invertible if and only if it does not have 0 as an eigenvalue: if is non-zero. But ThePerfectHacker's method is much better.