How did Frobenius prove the Cayley-Hamilton theorem for a general matrix? I heard Cayley did the case $n=2$ and then Hamilton generalized the result. How is this done? Assume it works for an $n\times n$ matrix and show $n+1?$
Edit: Sorry, Cayley showed it for $3\times 3$, I mean Frobenius.
 A: The concept of the minimal polynomial was introduced by Frobenius in his landmark 1878 paper. He used the language of bilinear forms and the theory of elementary divisors introduced by Weierstrass. He applied the minimal polynomial (called by him 'the degree of equation of the smallest degree $\varphi(A) = 0$') to give one of the first complete proofs of the Cayley-Hamilton theorem.

Hamilton had shown this result for quarternions and Cayley had proved it for matrices of orders $N=2$ and $N=3$. Frobenius however proved the theorem by showing that the minimal polynomial $\varphi$ of a matrix(or bilinear form) $A$ is a divisor of the characteristic polynomial $\psi$. Hence $\varphi(A) =0 \Rightarrow \psi(A) = 0$. 
You can also refer to "Frobenius and the symbolical algebra of matrices" in the Archive for History of Exact Sciences here. Hope it helps.
A: He didn't. He only proved it for three dimensional matrices. In fact, he shows that the matrix satisfies a polynomial but did not realize the polynomial was the characteristic one.
Similarly, Cayley only gave a proof for the case of 2 by 2 matrices — he identified the coefficients of the polynomial correctly. He says somewhere that he knows how to do the general case, but he does not even state what he means. This is in his Treatise on matrices.
