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Suppose that there is a matrix $A$.

Then $A$ is factorized into $A = LU$ form.

Why is $L$ always invertible? (or when is $L$ invertible?)

Also, Why is the reduced Echelon form of $U$ identity matrix when $A$ is invertible?

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$L$ is invertible because it is formed as a product of elementary matrices which are themselves invertible.

If $A$ is invertible then think about the equation $A=LU$. Is it possible for $U$ to be non-invertible? I'm not sure what you have to work with, if you have determinants it's simple to argue $det(U) \neq 0$. Or you could argue by the uniqueness of the homogeneous equation $Ax=0$ iff $x=0$. Once you know $U$ is invertible it follows $rref(U)=I$. However, I may be putting the cart before the horse in your situation. The argument you give should only use those tools your course has thus far discovered.

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