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

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

write B as a linear combination of the other matrices, if possible.
$B=\left[\begin{array}{cc}2& 3\\ -4& 2\end{array}\right],A_1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],A_2=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],A_3=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

Prove the following
A(rB)= r(AB)= (rA)B
where r and s are real numbers and A and B are matrices

If possible, find scalars $c_1,c_2,\text{ and }{c}_3$ so that
$c_1\left[\begin{array}{c}1\\ 2\\ -3\end{array}\right]+c_2\left[\begin{array}{c}-1\\ 1\\ 1\end{array}\right]+c_3\left[\begin{array}{c}-1\\ 4\\ -1\end{array}\right]=\left[\begin{array}{c}2\\ -2\\ 3\end{array}\right]$

Multiply the Following matrices:
$\left[\begin{array}{cc}2& 3\\ 1& 0\end{array}\right]\times \left[\begin{array}{cc}4& 1\\ 2& 1\end{array}\right]$
$\left[\begin{array}{cc}1& 3\\ 4& 5\\ 1& 2\end{array}\right]\times \left[\begin{array}{ccc}2& -1& 4\\ 3& 1& 0\end{array}\right]$

Explain why S is not a basis for $M_{2,2}$
$S=\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]\}$