Let Q be a 3 by 2 matrix having orthonormal columns Find det

Let Q be a 3 by 2 matrix having orthonormal columns. Find det(Q^TQ) and det (QQ^T).

Solution

Q^TQ is of order 2by2

QQ^T is of order 3by3

The matrix that projects onto the column space of Q is:P = Q^T(Q^TQ)¹Q^T

Given the columns of Q are orthonormal, then Q^TQ = I

det(Q^T Q) = det(I) = 1

det(QQ^T) = det(I) = 1