Question

Show that the determinant of a unitary matrix is a complex number with absolute value=1. Hint: See proof of equation (7.11).

Short answer

The determinant of the unitary matrix is unity.

Step-by-step solution

Given information.

The matrix is unitary and the absolute value is 1.

Unitary matrix.

A unitary matrix is one whose conjugate transpose equals its inverse. Real orthogonal matrices have a complex analogue in the form of unitary matrices.

If Matrix A satisfies the condition $\mathit{A}{\mathit{A}}^{\mathbf{†}}\mathbf{=}{\mathit{A}}^{\mathbf{†}}\mathit{A}\mathbf{=}\mathit{I}$, then Matrix Ais a unitary matrix.

Show that the determinant of a unitary matrix is a complex number with absolute value of unity

Consider U is a unitary matrix.

$$\begin{gathered} U^{-1}=U^T \\ U{U}^T=I \\ \det \left(U{U}^T\right)=\det I \\ \det \left(U\right)\det \left(U^T\right)=\det I(\because \det (AB)=(\det A\left)\right(\det B\left)\right) \end{gathered}$$

Since,

$$\begin{gathered} \det {\text{U}}^T=\det \left(U^{-1}\right) \\ \det {\text{U}}^T=\det \left(\overline{U}\right) \\ \det I=1 \end{gathered}$$

Then,$\left(\det U\right)\left(\det \overline{U}\right)=1$

Now,

$$\begin{gathered} z\overline{z}=1 (\because \det U=z) \\ |z{|}^2=1 \\ |z|=1 \end{gathered}$$

Hence, from the above equation, it is clear that the determinant of the unitary matrix is unity.