Thread: complex conjugates and unitary

For M M_nxn(C), let the be the matrix such that _ij= _ij for all i,j, where _ij is the complex conjugate of M_ij.

(1) Prove that det =

(2) A matrix Q in M_nxn(C)is called unitary if QQ*=I, where Q*= . Prove that if Q is a unitary matrix, then |det(Q)|=1

Originally Posted by tn11631

For M M_nxn(C), let the be the matrix such that _ij= _ij for all i,j, where _ij is the complex conjugate of M_ij.

(1) Prove that det =


This follows at once from the definition of determinant and because complex conjugation is and additive and multiplicative function


(2) A matrix Q in M_nxn(C)is called unitary if QQ*=I, where Q*= . Prove that if Q is a unitary matrix, then |det(Q)|=1

As ,as was proved in (1), you get ...etc.

Tonio