Page 190 - 35Linear Algebra

190 Determinants

8.4.1 Determinant of the Inverse

Let M and N be n × n matrices. We previously showed that

det(MN) = det M det N, and det I = 1.
Then 1 = det I = det(MM −1 ) = det M det M −1 . As such we have:

Theorem 8.4.1.
1
det M −1 =
det M

8.4.2 Adjoint of a Matrix

Recall that for a 2 × 2 matrix

d −b a b a b
= det I .
−c a c d c d
Or in a more careful notation: if

m m
1 1
M = 1 2 ,
m 2 1 m 2 2
then
2 1
1 m −m
M −1 = 2 2 2 1 ,
1
1
2
m m − m m 2 −m m
1 2 2 1 1 1
2 1
m −m
2
2
1
1
so long as det M = m m − m m 6= 0. The matrix 2 2 that
1
1
2
2
−m 2 1 m 1
1
appears above is a special matrix, called the adjoint of M. Let’s define the
adjoint for an n × n matrix.
i
The cofactor of M corresponding to the entry m of M is the product
j
i
i
of the minor associated to m and (−1) i+j . This is written cofactor(m ).
j
j
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