If and is supertriangular then
theorem: Let be commutative ring with identity.If an n-square matrix is supertriangular then .
proof: Find the characteristic polynomial of by computing the determinant
of . The square matrix
is a triangular matrix
. The determinant of a triangular matrix is the product of the diagonal element of the matrix. Therefore the characteristic polynomial is and by the Cayley-Hamilton theorem
the matrix satisfies the polynomial
. That is .
QED