# Property of an invertible matrix that row reduced form is identity matrix

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?

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 $n$).