530 CHAPTER 28. THE MATHEMATICAL THEORY OF DETERMINANTS∗

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”.

Proof: By Proposition 28.1.3 when two rows are switched, the determinant of the re-sulting matrix is (−1) times the determinant of the original matrix. By Corollary 28.1.5 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) .

If A has two equal columns or two equal rows, then switching them results in the samematrix. Therefore, det(A) =−det(A) and so det(A) = 0.

It remains to verify the last assertion.

det(A)≡ ∑(k1,··· ,kn)

sgn(k1, · · · ,kn)a1k1 · · ·(xaki + ybki

)· · ·ankn

= x ∑(k1,··· ,kn)

sgn(k1, · · · ,kn)a1k1 · · ·aki · · ·ankn

+y ∑(k1,··· ,kn)

sgn(k1, · · · ,kn)a1k1 · · ·bki · · ·ankn

≡ xdet(A1)+ ydet(A2) .

The same is true of columns because det(AT)= det(A) and the rows of AT are the columns

of A. ■

28.1.5 Linear Combinations And DeterminantsLinear combinations have been discussed already. However, here is a review and some newterminology.

Definition 28.1.7 A vector w, is a linear combination of the vectors {v1, · · · ,vr} if thereexists scalars, c1, · · ·cr such that w = ∑

rk=1 ckvk. This is the same as saying

w ∈ span(v1, · · · ,vr) .

The following corollary is also of great use.

Corollary 28.1.8 Suppose A is an n×n matrix and some column (row) is a linear combi-nation of r other columns (rows). Then det(A) = 0.

Proof: Let A =(

a1 · · · an

)be the columns of A and suppose the condition that

one column is a linear combination of r of the others is satisfied. Then by using Corollary