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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?

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The inverted matrix will have an identity row at the exact same position. Here is how you can know this. Write down a matrix equation with you matrix:

$$ \begin{matrix} \\ \text{$i$-th row} \to \\ \, \end{matrix} \begin{pmatrix} && \vdots \\ \hline 0&\cdots &1& \cdots&0\\ \hline && \vdots \end{pmatrix} \begin{pmatrix} \vdots \\x_i \\\vdots \end{pmatrix} = \begin{pmatrix} \vdots \\ y_i \\ \vdots \end{pmatrix}. $$

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.

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