# Inverting matrix with an identity row

By identity row, I mean a row in a square matrix that that has 0s everywhere except for a 1 in the n-th column. When you invert an NxN matrix with an identity row, does the inverted matrix always preserve the number of zeros (N-1) for that specific identity row?

$$\begin{matrix} \\ \text{i-th row} \to \\ \, \end{matrix} \begin{pmatrix} && \vdots \\ \hline 0&\cdots &1& \cdots&0\\ \hline && \vdots \end{pmatrix} \begin{pmatrix} \vdots \\x_i \\\vdots \end{pmatrix} = \begin{pmatrix} \vdots \\ y_i \\ \vdots \end{pmatrix}.$$
The $i$-th row written as isolated equation looks like $x_i=y_i$. Therefore, if we bring the matrix to the other side by inverting it, the $i$-th row must look the same to yield the same identity.