102 CHAPTER 3. DETERMINANTS

3. If you switch two rows or two columns, the determinant of the resulting matrix is −1times the determinant of the unswitched matrix. (This and the previous one say

(a1 · · ·an) → det (a1 · · ·an)

is an alternating multilinear function or alternating tensor.

4. det (e1, · · · , en) = 1.

5. det (AB) = det (A) det (B)

6. det (A) can be expanded along any row or any column and the same result is obtained.

7. det (A) = det(AT)

8. A−1 exists if and only if det (A) ̸= 0 and in this case(A−1

)ij=

1

det (A)cof (A)ji (3.13)

9. Determinant rank, row rank and column rank are all the same number for any m× nmatrix.

3.4 The Cayley Hamilton Theorem

Definition 3.4.1 Let A be an n× n matrix. The characteristic polynomial is defined 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−1t

n−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 3.4.2 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.

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