# Is it possible for a triangular matrix in echelon form to not have a unique solution and how?

I want to know if it is possible for a triangular matrix in echelon form to not have a unique solution and how?

Isn't there something to do with the determinant that shows this? or am I wrong?

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Note that a triangular matrix may have zeros on the diagonal, for example, $$\pmatrix{1&2&3\cr 0&4&5\cr0&0&0\cr}\ .$$ A system with this echelon form for its left hand side will have zero determinant, and will have infinitely many solutions or no solution, depending on the right hand side.