1.9. THE MATHEMATICAL THEORY OF DETERMINANTS 37

Observation 1.9.7 There are n! ordered lists of distinct numbers from

{1, · · · ,n}

To see this, consider n slots placed in order. There are n choices for the first slot. Foreach of these choices, there are n−1 choices for the second. Thus there are n(n−1) waysto fill the first two slots. Then for each of these ways there are n−2 choices left for the thirdslot. Continuing this way, there are n! ordered lists of distinct numbers from {1, · · · ,n} asstated in the observation.

1.9.3 A Symmetric DefinitionWith the above, it is possible to give a more symmetric description of the determinant fromwhich it will follow that det(A) = det

(AT).

Corollary 1.9.8 The following formula for det(A) is valid.

det(A) =1n!· ∑(r1,··· ,rn)

∑(k1,··· ,kn)

sgn(r1, · · · ,rn)sgn(k1, · · · ,kn)ar1k1 · · ·arnkn . (1.23)

And also det(AT)= det(A) where AT is the transpose of A. (Recall that for AT =

(aT

i j

),

aTi j = a ji.)

Proof: From Proposition 1.9.6, if the ri are distinct,

det(A) = ∑(k1,··· ,kn)

sgn(r1, · · · ,rn)sgn(k1, · · · ,kn)ar1k1 · · ·arnkn .

Summing over all ordered lists, (r1, · · · ,rn) where the ri are distinct, (If the ri are notdistinct, sgn(r1, · · · ,rn) = 0 and so there is no contribution to the sum.)

n!det(A) = ∑(r1,··· ,rn)

∑(k1,··· ,kn)

sgn(r1, · · · ,rn)sgn(k1, · · · ,kn)ar1k1 · · ·arnkn .

This proves the corollary since the formula gives the same number for A as it does for AT .■

Corollary 1.9.9 If two rows or two columns in an n× n matrix A, are switched, thedeterminant of the resulting matrix equals (−1) times the determinant of the original ma-trix. If A is an n× n matrix in which two rows are equal or two columns are equal thendet(A) = 0. Suppose the ith row of A equals

(xa1 + yb1, · · · ,xan + ybn)

Thendet(A) = xdet(A1)+ ydet(A2)

where the ith row of A1 is (a1, · · · ,an) and the ith row of A2 is (b1, · · · ,bn) , all other rows ofA1 and A2 coinciding with those of A. In other words, det is a linear function of each rowA. The same is true with the word “row” replaced with the word “column”.