Please help with problem 5 part A thanks let uvw be vectors

Please help with problem 5 part A thanks

\"let u,v,w be vectors in R4 and define the transformation T: R4-->R4 by the rule T(x)=det([x u v w]) prove that there exists a vector z in R4 such that T(x)=z(dot)x for all x in R4 and find the components of z interms of the vectors u, v, w

Prove that if A is a n times n matrix with integer entries, then det(>l) is an integer. Prove that if A is a n times n matrix with integer entries, then A-^1 has integer entries if and only if det(i4) = plusminus 1. (you can use the classical adjoint of A.) in 5. Let u, v, w be vectors in R^4, and define the transformation T : R \' -? R by the rule

Solution

Solution : 4 ( b )

Proof. The fact that det(A) = +/- 1 implies that when we perform Gaussian elimination on A , we never have to multiply rows by scalars. This means that for each column, the pivot entry is created by the previous column\'s row operations and can be brought into place by swapping rows. (And the first column must already contain a 1.) Therefore, we never need to multiply by a non-integral value to perform Gaussian elimination.