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Is there a way to know if some non-sqaure matrix will have the identity matrix in the left side of its reduced row echelon form without doing the elementary row operations?

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Note that it is only possible for the rref of a matrix $A$ to have the identity matrix on the left side (i.e. to be of the form $[I \quad M]$) if $A$ is $m \times n$ with $m \leq n$. That is, $A$ must be a "wide" matrix.

Given that $A$ is wide, there will be an identity matrix on the left side of the rref if and only if the first $m$ columns of $A$ are linearly independent. In other words, if and only if the matrix formed from the first $m$ columns of $A$ is invertible.

In some cases there might be shortcuts that one could use (for instance if $A$ is upper triangular), but in general the most efficient way to check that a matrix is invertible by hand is to use row operations.

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