## Quantum Country exercise 5

This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 5. Show that the matrix $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ is unitary.

Answer: A matrix $U$ is unitary if $U^\dagger U = I$ where $U^\dagger$ is the adjoint matrix to $U$ (produced by taking the transpose $U^T$ of $U$ and then replacing all values by their complex conjugates) and $I$ is the identity matrix.

Since $X$ is symmetric we have $X^T = X$ and since the values of $X$ (and thus of $X^T$) are all real we have $\left( X^T \right)^* = X^T$. We thus have

$X^\dagger X = \left( X^T \right)^* X = X^T X = XX$

$= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$

Since $X^\dagger X = I$ the matrix $X$ is unitary.

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