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The sum of two lower triangular matrices is always
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The correct answer is:
a lower triangular matrix
Let lower triangular matrix
$A=\left[\begin{array}{lll}a & 0 & 0 \\ p & b & 0 \\ q & r & c\end{array}\right]$ and $B=\left[\begin{array}{ccc}a_1 & 0 & 0 \\ p_1 & b_1 & 0 \\ q_1 & r_1 & c_1\end{array}\right]$
$A+B=\left[\begin{array}{ccc}a+a_1 & 0 & 0 \\ p+p_1 & b+b_1 & 0 \\ q+q_1 & r+r_1 & c+c_1\end{array}\right]$
Clearly $A+B$ is also lower triangular matrix.
$A=\left[\begin{array}{lll}a & 0 & 0 \\ p & b & 0 \\ q & r & c\end{array}\right]$ and $B=\left[\begin{array}{ccc}a_1 & 0 & 0 \\ p_1 & b_1 & 0 \\ q_1 & r_1 & c_1\end{array}\right]$
$A+B=\left[\begin{array}{ccc}a+a_1 & 0 & 0 \\ p+p_1 & b+b_1 & 0 \\ q+q_1 & r+r_1 & c+c_1\end{array}\right]$
Clearly $A+B$ is also lower triangular matrix.
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