Can we find a unitary matrix to remove entries?

Question

Does there exists a $3\times 3$ unitary matrix $U$ such $$ U\begin{bmatrix} 1 & 0 & a\\ 0 & 1 & b\\ 0 & 0 & 1 \end{bmatrix} $$ is either $\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & x\\ 0 & 0 & 1 \end{bmatrix}$ or $\begin{bmatrix} 1 & 0 & y\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$ where $x\neq b$ and $y\neq a$ are complex numbers.(Before this question was edited, the condition was that $x=b$ and $y=a$.)

Answer 1

Multiplication by a unitary matrix preserves the norm of a vector. The last column of $UB$ is $U$ times the last column of $B$, and for a unitary matrix you can't have $U \pmatrix{a\cr b \cr 1} = \pmatrix{0 \cr b \cr 1}$ unless $a=0$, or $\pmatrix{a \cr 0 \cr 1}$ unless $b = 0$.