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If a the row reduced form of a $n \times n$ matrix is the equivalent $n \times n$ identity matrix. Is the $n \times n$ matrix always invertible?

Furthermore if the row reduced form is not the equivalent $n \times n$ identity matrix is the $n \times n$ matrix not invertible?

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What you said is true. Being invertible is equivalent to a huge list of other properties one of which is that the matrix is full rank (in the case of an $n\times n$ matrix, this means that the rank of the matrix is $n$).

A full rank square matrix has the identity matrix as its reduced row echelon form.

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  • $\begingroup$ Note that the fact that our scalars come from a field is important here. This is not true, for instance, for matrices with integer entries. $\endgroup$ – Ken Duna Jun 20 '16 at 22:19

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