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

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Matrices asked 2021-03-05

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

## Answers (1) 2021-03-06

Consider 2 x 2 lower triangular matrix

$$\displaystyle{A}={\left[\begin{matrix}{a}&{0}\\{b}&{c}\end{matrix}\right]}{\quad\text{and}\quad}{B}={\left[\begin{matrix}{d}&{0}\\{e}&{f}\end{matrix}\right]}$$

Now taking multiplication of A and B we get,

$$\displaystyle{A}{B}={\left[\begin{matrix}{a}&{0}\\{b}&{c}\end{matrix}\right]}{\left[\begin{matrix}{d}&{0}\\{e}&{f}\end{matrix}\right]}$$

$$\displaystyle{A}{B}={\left[\begin{matrix}{a}{d}&{0}\\{b}{d}+{c}{e}&{c}{f}\end{matrix}\right]}$$

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.

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