Prove that if A^T A = In, then detA = ±1.

Question

Prove that if A^T A = In, then detA = ±1.

Answer 1

There are a few facts about determinants you will need here.

 

1) det(A) = det(A^T)

2) det(AB) = det(A) det(B)

3) det(I_n) = 1 for any positive integer, n.

 

Proof:

det(A^TA) = det(A^T) det(A) = [det(A)]² and det(I_n) = 1.

 

Thus,

 

[det(A)]² = 1, so after taking square roots, you have the equation det(A) = ±1.

 

This completes the proof.