Will the product of these two matrices be unitary?

Question

$U$ is a unitary matrix of order $m$.

$V$ is a unitary matrix of order $m-1$.

Let $V’$ be the a matrix of order $m$, constructed as $\left( \begin{array}{cc} 1 & 0 \\ 0 & V \end{array} \right)$.

Is $UV’$ also unitary?

Answer 1

$V'$ is unitary, therefore yes.