linear-guest

8.4 Properties of the Determinant 191
Definition For M = (m i j) a square matrix, the adjoint matrix adj M is
given by
adj M = (cofactor(m i j)) T .
Example 107
⎛ ( ) 2 0
det
⎛ ⎞
1 1
3 −1 −1
( )
adj ⎝1 2 0⎠ =
−1 −1
− det
0 1 1
1 1
⎜ ( )
⎝ −1 −1
det
2 0
( ) 1 0
− det
0 1
( ) 3 −1
det
0 1
( ) 3 −1
− det
1 0
( ) ⎞
1 2
det
0 1
( )
3 −1
− det
0 1
( ) ⎟ 3 −1 ⎠
det
1 2
T
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 ≠ 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 ≠ 0, the adjoint gives an explicit formula for M −1 .
Theorem 8.4.2. For M a square matrix with det M ≠ 0 (equivalently, if M
is invertible), then
M −1 = 1
det M adj M
The Adjoint Matrix
191