442 CHAPTER 20. THEORY OF DETERMINANTS∗

20.2.10 The Cayley Hamilton Theorem

Definition 20.2.20 Let A be an n × n matrix. The characteristic polynomial isdefined as

qA (t)≡ det(tI −A)

and the solutions to qA (t) = 0 are called eigenvalues. For A a matrix and p(t) = tn +an−1tn−1 + · · ·+a1t +a0, denote by p(A) the matrix defined by

p(A)≡ An +an−1An−1 + · · ·+a1A+a0I.

The explanation for the last term is that A0 is interpreted as I, the identity matrix.

The Cayley Hamilton theorem states that every matrix satisfies its characteristic equa-tion, that equation defined by qA (t) = 0. It is one of the most important theorems in linearalgebra1. The proof in this section is not the most general proof, but works well when thefield of scalars is R or C. The following lemma will help with its proof.

Lemma 20.2.21 Suppose for all |λ | large enough,

A0 +A1λ + · · ·+Amλm = 0,

where the Ai are n×n matrices. Then each Ai = 0.

Proof: Suppose some Ai ̸= 0. Let p be the largest index of those which are non zero.Then multiply by λ

−p.

A0λ−p +A1λ

−p+1 + · · ·+Ap−1λ−1 +Ap = 0

Now let λ → ∞. Thus Ap = 0 after all. Hence each Ai = 0.With the lemma, here is a simple corollary.

Corollary 20.2.22 Let Ai and Bi be n×n matrices and suppose

A0 +A1λ + · · ·+Amλm = B0 +B1λ + · · ·+Bmλ

m

for all |λ | large enough. Then Ai = Bi for all i. If Ai = Bi for each Ai,Bi then one cansubstitute an n×n matrix M for λ and the identity will continue to hold.

Proof: Subtract and use the result of the lemma. The last claim is obvious by matchingterms.

With this preparation, here is a relatively easy proof of the Cayley Hamilton theorem.

Theorem 20.2.23 Let A be an n × n matrix and let q(λ ) ≡ det(λ I −A) be thecharacteristic polynomial. Then q(A) = 0.

Proof: Let C (λ ) equal the transpose of the cofactor matrix of (λ I −A) for |λ | large.(If |λ | is large enough, then λ cannot be in the finite list of eigenvalues of A and so for suchλ , (λ I −A)−1 exists.) Therefore, by Theorem 20.2.18

C (λ ) = q(λ )(λ I −A)−1 .

1A special case was first proved by Hamilton in 1853. The general case was announced by Cayley some timelater and a proof was given by Frobenius in 1878.