Tooth decay occurs when tooth enamel wears away over time. It exponential decay models the change in the number of dentists in the United States over

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.

Help with exact value of:
${\displaystyle \tan ({\sin }^{-1}(-\frac{1}{2})-{\tan }^{-1}\left(\frac{3}{4}\right))}$

How can one prove:
${\displaystyle \cos \left(\frac{\pi }{7}\right)+\cos \left(3\frac{\pi }{7}\right)+\cos \left(5\frac{\pi }{7}\right)=\frac{1}{2}}$

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

Proving
${\displaystyle \sin \left((n-\frac{1}{2})\varphi \right)+\sin \left(\frac{\varphi }{2}\right)=\sin \left(\frac{n+1}{2}\varphi \right)}$

Factor the following polynomials.
a) ${\displaystyle x^2-8x}$
b) ${\displaystyle x^3+3x^2-18x}$
c) ${\displaystyle x^3-10x}$