508 CHAPTER 27. DETERMINANTS

Summarizing,n

∑i=1

air cof(A)ik det(A)−1 = δ rk ≡

{1 if r = k0 if r ̸= k

.

Nown

∑i=1

air cof(A)ik =n

∑i=1

air cof(A)Tki

which is the krth entry of cof(A)T A. Therefore,

cof(A)T

det(A)A = I. (27.2)

Using the other formula in Definition 27.1.13, and similar reasoning,

n

∑j=1

ar j cof(A)k j det(A)−1 = δ rk

Nown

∑j=1

ar j cof(A)k j =n

∑j=1

ar j cof(A)Tjk

which is the rkth entry of Acof(A)T . Therefore,

Acof(A)T

det(A)= I, (27.3)

and it follows from 27.2 and 27.3 that A−1 =(

a−1i j

), where

a−1i j = cof(A) ji det(A)−1 .

In other words,

A−1 =cof(A)T

det(A).

Now suppose A−1 exists. Then by Theorem 27.1.26,

1 = det(I) = det(AA−1)= det(A)det

(A−1)

so det(A) ̸= 0. ■This way of finding inverses is especially useful in the case where it is desired to find

the inverse of a matrix whose entries are functions.

Example 27.2.4 Suppose

A(t) =

 et 0 00 cos t sin t0 −sin t cos t

Show that A(t)−1 exists and then find it.