# Product of triangular and symmetric matrices is triangular?

I have a product

$\matrix{U}$ $\matrix{S}$ $\matrix{L}$

with $\matrix{U}$ upper triangular, $\matrix{L}$ lower triangular and $\matrix{S}$ symmetric. Is the resulting matrix still triangular?

No. A counterexample is $U = L = I$ (identity matrix) and $S =$ the square matrix having all entries equal to $1$.