matrix Q is called orthogonal if Q1QT Show that if Q is orth

matrix Q is called orthogonal if Q1=QT. Show that if Q is orthogonal, qij=Qij/det(Q)

Solution

If the matrix Q is orthogonal, then Q^T = Q = Q^-1.

We know that for any invertible matrix Q, we have Q^-1 = 1/detQ.AdjQ = [1/det Q](C^T), where C is the cofactor matrix of Q.Now, since Q^-1 = Q, we have, Q = [1/det Q](C^T).

Further, since Q = Q^T,  we have Q^T = [1/det Q](C^T) . Therefore, Q = [1/det Q](C)

Thefore, q_ij  = Q_ij / detQ ( as detQ is a scalar and the component/element of C corresponding to q_ij in Q is Q_ij in C)