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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?

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

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