Question: Prove that the determinant can be computed by cofactor expansion along any row or column. More precisely, if A Mn(F) with n 2, prove that

Prove that the determinant can be computed by cofactor expansion along any row or column. More precisely, if A Mn(F) with n 2, prove that

(1) detA =

n SUM i=1 of (1)i+j det(Ai,j)ai,j for all 1 j n,

and that

(2) detA =

n SUM j=1 of (1)i+j det(Ai,j)ai,j for all 1 i n.

Hint: To prove (1), let B be the matrix obtained from A by interchanging the rst and j-th columns.

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