The diagram below shows 3 vectors which sum to zero, all ofequal length.

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

How to prove $\sum _{k=1}^n\cos (\frac{2\pi k}{n})=0$ for any n>1?

Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.
$\sum _{n=2}^{\mathrm{\infty }}\frac{1}{4^n}$

Please, rewrite the expression as an algebraic expression in x.
$\sin ({\tan }^{-}1x)$

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

Prove trigonometric equivalence
${\displaystyle \sin x+\sin y=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)}$