On the Adjugate of a Matrix

Dn−1 = I −A · Dn−1 + Dn−2 = cn−1 I −A · Dn−2 + Dn−3 = cn−2 I .. .. .. −A · D1 + D0 = c1 I −A · D0 = c0 I Finally, by multiplying the first equation by An, the second by A, and so on, up to the last equation, and by adding the new equations up, we obtain the Cayley-Hamilton theorem. This is the proof in [1, p. 50]. If, instead, we multiply the first equation by A, the second by A, etc., and stop precisely before the last equation, by summing up we obtain at the left-hand side D0, which is (−1) A, say, by putting λ = 0. Hence, we get (1). (2) can be proven by obtaining the coefficients of (λ I −A) adj step by step through the same procedure.