Is this matrix equal to the identity matrix

Question

I'm trying to prove if this matrix is unitary:
$\begin{bmatrix}0 & -i \\i & 0 \end{bmatrix}$
So after multiplying it by it's conjugate transpose I got the answer $\begin{bmatrix}-i & 0 \\0 &-i \end{bmatrix}$
Is this equal to the identity matrix?

Answer 1

A matrix is unitary if its conjugate transpose is also its inverse.

Call your first matrix $A$. The conjugate transpose of your first matrix is $ \bar{A^t} = \begin{bmatrix} 0 &i \\ -i &0 \end{bmatrix}^T = \begin{bmatrix} 0 & -i \\ i &0 \end{bmatrix}$ So you can see that your matrix is equal to its conjugate transpose. That property is called Hermitian. Have you tried multiplying $ A \bar{A^t} = \begin{bmatrix} 0 & -i \\ i &0 \end{bmatrix}^2 = \ \ ?$