Page 189 - 35Linear Algebra

8.4 Properties of the Determinant 189

Figure 8.7: Transposes leave the determinant unchanged.

Then we can write out the above in formulas as follows:

X
det M = sgn(σ)m 1 m 2 · · · m n
σ(1) σ(2) σ(n)
σ
X σ −1 (1) σ −1 (2) −1
= sgn(σ)m 1 m 2 · · · m σ (n)
n
σ
X σ −1 (1) σ −1 (2) −1
−1
= sgn(σ )m 1 m 2 · · · m σ (n)
n
σ
X σ(1) σ(2)
= sgn(σ)m m · · · m σ(n)
1 2 n
σ
T
= det M .
The second-to-last equality is due to the existence of a unique inverse permu-
tation: summing over permutations is the same as summing over all inverses
of permutations (see review problem 3). The final equality is by the definition
of the transpose.

Example 106 Because of this, we see that expansion by minors also works over
columns. Let
 
1 2 3
M =  0 5 6  .
0 8 9
Then

5 8
T
det M = det M = 1 det = −3 .
6 9
189