118 CHAPTER 6. DETERMINANTS

In other words,

A−1 =cof(A)T

det(A).

Now suppose A−1 exists. Then by Theorem 6.1.26,

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. It also has enormous theoretical signif-icance in more advanced mathematics.

Example 6.2.4 Suppose

A(t) =

 et 0 00 cos t sin t0 −sin t cos t

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

First note det(A(t)) = et ̸= 0 so A(t)−1 exists. The cofactor matrix is

C (t) =

 1 0 00 et cos t et sin t0 −et sin t et cos t

and so the inverse is

1et

 1 0 00 et cos t et sin t0 −et sin t et cos t

T

=

 e−t 0 00 cos t −sin t0 sin t cos t

 .

6.2.2 Cramer’s RuleThis formula for the inverse also implies a famous procedure known as Cramer’s rule.Cramer’s rule gives a formula for the solutions, x, to a system of equations, Ax = y in thespecial case that A is a square matrix. Note this rule does not apply if you have a system ofequations in which there is a different number of equations than variables.

In case you are solving a system of equations, Ax = y for x, it follows that if A−1 exists,

x =(A−1A

)x = A−1 (Ax) = A−1y

thus solving the system. Now in the case that A−1 exists, there is a formula for A−1 givenabove. Using this formula,

xi =n

∑j=1

a−1i j y j =

n

∑j=1

1det(A)

cof(A) ji y j.