22.1.4 The Alternating Property Of The Determinant
Corollary 22.1.6If two rows or two columns in an n×n matrix A, are switched, the determinant of theresulting matrix equals
(− 1)
times the determinant of the original matrix. If A is an n×n matrix in whichtwo rows are equal or two columns are equal then det
(A )
= 0. Suppose the ithrow of A equals
(xa1 + yb1,⋅⋅⋅,xan + ybn)
. Then
det(A ) = xdet(A1)+ y det(A2 )
where the ithrow of A1is
(a1,⋅⋅⋅,an)
and the ithrow of A2is
(b1,⋅⋅⋅,bn)
, all other rows of A1and A2coinciding with those of A. In other words, det is a linear function of each row A. The same is true withthe word “row” replaced with the word “column”.
Proof: By Proposition 22.1.3 when two rows are switched, the determinant of the resulting matrix is
(− 1)
times the determinant of the original matrix. By Corollary 22.1.5 the same holds for columns because the
columns of the matrix equal the rows of the transposed matrix. Thus if A1 is the matrix obtained from A
by switching two columns,
det(A ) = det(AT) = − det(AT ) = − det(A ).
1 1
If A has two equal columns or two equal rows, then switching them results in the same matrix. Therefore,
det