Page 191 - 35Linear Algebra

8.4 Properties of the Determinant 191

i
Definition For M = (m ) a square matrix, the adjoint matrix adj M is
j
given by
i
T
adj M = (cofactor(m )) .
j
Example 107
 T

2 0 1 0 1 2
det − det det
1 1 0 1 0 1
 
   
3 −1 −1  
 −1 −1 3 −1 3 −1 
adj  1 2 0  =  − det det − det 
 1 1 0 1 0 1 
0 1 1  
 
−1 −1 3 −1
 3 −1 
det − det det
2 0 1 0 1 2
Reading homework: problem 6
Let’s compute the product M adj M. For any matrix N, the i, j entry
of MN is given by taking the dot product of the ith row of M and the jth
column of N. Notice that the dot product of the ith row of M and the ith
column of adj M is just the expansion by minors of det M in the ith row.
Further, notice that the dot product of the ith row of M and the jth column
of adj M with j 6= i is the same as expanding M by minors, but with the
jth row replaced by the ith row. Since the determinant of any matrix with
a row repeated is zero, then these dot products are zero as well.
We know that the i, j entry of the product of two matrices is the dot
product of the ith row of the first by the jth column of the second. Then:

M adj M = (det M)I

Thus, when det M 6= 0, the adjoint gives an explicit formula for M −1 .

Theorem 8.4.2. For M a square matrix with det M 6= 0 (equivalently, if M
is invertible), then
1
M −1 = adj M
det M

The Adjoint Matrix

191