I have two $2N\times 2N$ matrices, defined by blocks:

$$ A = \begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix} $$ $$ B = \begin{bmatrix} 0 & 0 \\ 0 & b \end{bmatrix} $$

where $a$ and $b$ are $N\times N$ matrices. I assume that $A\ne B$, i.e. they are not trivially 0.

Then I have a coordinate transformation defined by a generic invertible matrix $T$. Through this transformation $A$ becomes $A'$ and $B$ becomes $B'$:

$$ A' = T^{-1} A T $$ $$ B' = T^{-1} B T $$

Of course, it is not possible that $A'=B'$ (proof: this would imply $A=B$)

Instead my question is if it is possible that the rows of $A'$ and $B'$ are equal, except one:

$$ \left[A'\right]_{i,j} = \left[B'\right]_{i,j} \tag{1} $$

for $i=1\dots 2N-1$ and $j=1\dots 2N$ (here $i$ the subscript of the row and $j$ of the column).

I guess that this cannot happen, unless $A$ and $B$ are trivially 0. But I could not prove it, although it does not seem too difficult.