Construct a 3\times3 matrix A, with nonzero entries, and a vector b in

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.

$(-5)\times \cdots \times (-5)17times$

55 athletes are running race. A gold medal is to be given to the winer, a silver medal is to be given to the second-place finisher, and bronze medal is to be given to the third-place finisher. Asume that there are no ties. In how many possible ways can the 3 medals be distributed

Find all z such that $|\tan z|=1$

We want to find $O(x,y,z)$ by using two known points $P(2,3,5)$ and $Q(1,2,2)$ and angle of $POQ=60$ degrees with plane $E:1.15714286x+1.8547619y-z-2.86101191=0$ and length $OP=OQ$
Is there any way to find it?

The reduced row echelon form of the matrix A is
$\left[\begin{array}{ccc}1& 2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$
Find three different such matrices A. Explain how you determined your matrices.