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

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.

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

For which values of the constants c and d is $\left[\begin{array}{c}5\\ 7\\ c\\ d\end{array}\right]$ a linear combination of $\left[\begin{array}{c}1\\ 1\\ 1\\ 1\end{array}\right]\text{ and }\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]$

Given the matrices
$A=\left[\begin{array}{cc}-1& 3\\ 2& -1\\ 3& 1\end{array}\right]\text{ and }B=\left[\begin{array}{cc}0& -2\\ 1& 3\\ 4& -3\end{array}\right]$ find the $3\times 2$ matrix X that is a solution of the equation. 8X+A=B

Let $A=\left[\begin{array}{ccc}1& 0& -2\\ -2& 1& 6\\ 3& -2& -5\end{array}\right]$ and $b=\left[\begin{array}{c}-1\\ 7\\ -3\end{array}\right]$. Define a linear transformation T by T(x)=AX. Determine a vector x whose image under T is b. Is the vector x that you found unique or not? Explain your answer.

Solve for x and y $\left[\begin{array}{cc}x& 2y\\ 4& 6\end{array}\right]=\left[\begin{array}{cc}2& -2\\ 2x& -6y\end{array}\right]$
$3\left[\begin{array}{cc}x& y\\ y& x\end{array}\right]=\left[\begin{array}{cc}6& -9\\ -9& 6\end{array}\right]$
$2\left[\begin{array}{cc}x& y\\ x+y& x-y\end{array}\right]=\left[\begin{array}{cc}2& -4\\ -2& 6\end{array}\right]$
$\left[\begin{array}{cc}x& y\\ -y& x\end{array}\right]-\left[\begin{array}{cc}y& x\\ x& -y\end{array}\right]=\left[\begin{array}{cc}4& -4\\ -6& 6\end{array}\right]$

For the matrices, list the eigenvalues, repeated according to their multiplicities.
$\left[\begin{array}{cccc}4& -7& 0& 2\\ 0& 3& -4& 6\\ 0& 0& 3& -8\\ 0& 0& 0& 1\end{array}\right]$