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I've been told that a matrix cannot be inverted if its row-reduced echelon matrix is not equal to the identity matrix. This is obvious to me in the sense that it cannot then be a valid solution to Ax = b, but other than that, is there any other reason behind this?

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Because its row rank is not the maximal rank (the dimension of the matrix), hence its column rank is not either, so its column vectors are not a basis of the vector space.

Because its determinant is $0$.

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