Find all 2 by 2 matrices that are orthogonal and also symmetric. Which two numbers can be eigenvalues of those two matrices?

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.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Determine the null space of each of the following matrices:
$\left[\begin{array}{cccc}1& 1& -1& 2\\ 2& 2& -3& 1\\ -1& -1& 0& -5\end{array}\right]$

Write each of the following linear combinations of columns as a linear system of the form in (4):
$a)x_1\left[\begin{array}{c}2\\ 0\end{array}\right]+x_2\left[\begin{array}{c}1\\ 3\end{array}\right]=\left[\begin{array}{c}4\\ 2\end{array}\right]$
$b)x_1\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]+x_2\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right]+x_3\left[\begin{array}{c}3\\ 4\\ 5\end{array}\right]+x_4\left[\begin{array}{c}1\\ 3\\ 4\end{array}\right]=\left[\begin{array}{c}2\\ 5\\ 8\end{array}\right]$

Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
${\displaystyle 2x_1-5x_2+8x_3=0}$
${\displaystyle -2x_1-7x_2+x_3=0}$
${\displaystyle 4x_1+2x_2+7x_3=0}$

Find x and y.
$\left[\begin{array}{cc}x& -2\\ 7& y\end{array}\right]=\left[\begin{array}{cc}-4& -2\\ 7& 22\end{array}\right]$

Explain why the formula is not valid for matrices. Illustrate your argument with examples.
$(A+B)(A+B)=A^2+2AB+B^2$