20.2. THE DETERMINANT 435

20.2.3 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 20.2.5 The following formula for det(A) is valid.

det(A) =1n!·

∑(r1,··· ,rn)

∑(k1,··· ,kn)

sgn(r1, · · · ,rn)sgn(k1, · · · ,kn)ar1k1 · · ·arnkn . (20.9)

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 20.2.3, 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 .

20.2.4 The Alternating Property of the Determinant

Corollary 20.2.6 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). Then

det(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”.

Proof: By Proposition 20.2.3 when two rows are switched, the determinant of the re-sulting matrix is (−1) times the determinant of the original matrix. By Corollary 20.2.5 thesame holds for columns because the columns of the matrix equal the rows of the transposedmatrix. Thus if A1 is the matrix obtained from A by switching two columns,

det(A) = det(AT )=−det

(AT

1)=−det(A1) .