Is their a minimum number of row operation for any nxn invertible matrix that turns it into the identity matrix? Is their a minimum number of row operation for any nxn invertible matrix that turns it into the identity matrix.
How would you go about showing the minimum number of row operation need to turn any nxn inveritble matrix into the identity matrix
 A: I think you can row reduce any $n\times n$ invertible matrix to the identity with at most $n^2$ elementary row operations.
Let's first focus on the number of steps need to 'clear a column' (i.e., get a $1$ in the $k$th position of column $k$ and $0$s everywhere else in column $k$).
There are two cases for clearing column $k$:
Case 1: There is a nonzero entry in the $k$th position of the column.  Then you will need (possibly) a row multiplication step to change the entry to a $1$. You will then need (up to) $n-1$ steps of adding a multiple of a row to a different row to clear the column.  [Total of (up to) $n$ elementary row operations.]
Case 2: There is a $0$ in the $k$th position of the column.  Because the matrix is invertible, there will be a nonzero entry in the $k$th column somewhere below the $k$th entry of the column.  So you will need a row swap. You then may need a row multiplication to get a $1$ in position.  But now to clear the column, you need at most $n-2$ steps to change the remaining column entries to $0$s (one entry is already a $0$ since you had to swap it out).  [So again: Total of (up to) $n$ elementary row operations.]
Since each column takes up to $n$ operations to clear, the whole matrix can be cleared with at most $n^2$ elementary row operations.
