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

Ayaana Buck 2020-11-30 Answered
True or False
If A and B are $n×n$ lower triangular matrices, then AB is also lower triangular
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Step 1
Given,
A and B are nxn lower triangular matrices.
Consider $2×2$ lower triangular matrix

Now taking multiplication of A and B we get,
$AB=\left[\begin{array}{cc}a& 0\\ b& c\end{array}\right]\left[\begin{array}{cc}d& 0\\ e& f\end{array}\right]$
$AB=\left[\begin{array}{cc}ad& 0\\ bd+ce& cf\end{array}\right]$
Step 2
Therefore,
AB is also an upper triangular matrix,
Hence,
If A and B are $n×n$ lower triangular matrices, then AB is also lower triangular
Therefore,
Given statement is true.

Jeffrey Jordon