True or FalseIf A and B are nxxn lower triangular matrices, then AB is also lower triangular

True or False If A and B are $n×n$ lower triangular matrices, then AB is also lower triangular

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Consider 2 x 2 lower triangular matrix

$A=\left[\begin{array}{cc}a& 0\\ b& c\end{array}\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{cc}d& 0\\ e& f\end{array}\right]$

Now taking multiplication of A and B we get,

$AB=\left[\begin{array}{cc}a& 0\\ b& c\end{array}\right]\left[\begin{array}{cc}d& 0\\ e& f\end{array}\right]$

$AB=\left[\begin{array}{cc}ad& 0\\ bd+ce& cf\end{array}\right]$

Therefore, AB is also an upper triangular matrix, N

Hence, if A and B are n x n lower triangular matrices, then AB is also lower triangular. Therefore, given statement is true.