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Abstract. The weighted mean matrix 
${{M}_{a}}$ is the triangular matrix 
$\left\{ {{a}_{k}}/{{A}_{n}} \right\}$, where 
${{a}_{n}}\,>\,0$ and 
${{A}_{n}}\,:=\,{{a}_{1}}\,+\,{{a}_{2}}\,+\cdots +\,{{a}_{n}}$. It is proved that, subject to 
${{n}^{c}}{{a}_{n}}$ being eventually monotonic for each constant 
$c$ and to the existence of 
$\alpha \,:=\,\lim \,\frac{{{A}_{n}}}{n{{a}_{n}}},\,{{M}_{a}}\,\in \,B\left( {{l}_{p}} \right)$ for 
$1\,<\,p\,<\infty $ if and only if 
$\alpha \,<\,p$.
The two theorems proved yield simple yet reasonably general conditions for triangular matrices to be bounded operators on 
${{l}_{p}}$. The theorems are applied to Nörlund and weighted mean matrices.
