Unit lower triangular matrices multiplication

Question

We know that product of two unit lower triangular matrices is a unit lower triangular matrix. However, if product of two lower triangular matrices is unit lower triangular then is it necessary for the constituent matrices of the product to be unit lower triangular? Is this an if and only if condition?

Answer 1

Not quite.

Let $C=AB$, where $A,B$ are $n{\times}n$ lower triangular matrices.

Then $C$ is lower triangular and it's easily verified that for $1\le i\le n$ we have $$C_{i,i}=A_{i,i}B_{i,i}$$ hence $C$ is unit lower triangular if and only if corresponding diagonal entries of $A,B$ are reciprocals of each other.