Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

If $A_{\alpha }=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)$, then show that $A_{{60}^{\circ }}·A_{-{30}^{\circ }}=A_{{30}^{\circ }}$.

If $\mathrm{A}=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$ is identity matrix, find $\alpha$.

If $\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\end{array}\right]\left[\begin{array}{l}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}5\end{array}\right]=\left[\begin{array}{l}7\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}11\\ k\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}23\end{array}\right]$ then write the value of $k$.

If $\mathrm{A}=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)$ and $\mathrm{A}+{\mathrm{A}}^{\mathrm{T}}=\mathrm{I}$, if the value of $\alpha$ is of the form of $\frac{\mathrm{\pi }}{a}$, then $a=$

Define ${\mathrm{Q}}_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$, prove that ${\mathrm{Q}}_{\theta }^2={\mathrm{Q}}_{2\theta }$.

Define ${\mathrm{Q}}_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$, prove that ${\mathrm{Q}}_{\theta }{\mathrm{Q}}_{\psi }={\mathrm{Q}}_{\theta +\psi }$.

If $\mathrm{P}=\left(\begin{array}{cc}{\cos }^2 \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & {\sin }^2 \theta \end{array}\right)$ show that ${\mathrm{P}}^2=\mathrm{P}$. Defining $\mathrm{H}=2\mathrm{P}-{\mathrm{I}}_2$ prove that $\mathrm{H}={\mathrm{H}}^{-1}$.

[Hints: Show ${\mathrm{H}}^2={\mathrm{I}}_2$ by using ${\mathrm{P}}^2=\mathrm{P}$]

In an election, a political party hired a public relations company to promote its candidate in three ways: SMS, telephone and letters. The cost per contact (in Rs.) is given by the matrix $A$ as follows:

$A=\begin{array}{cc}0.2& \mathrm{sms}\\ 0.6& \mathrm{telephone}\\ 2& \mathrm{letter}\end{array}$

The number of contacts of each type made in two cities $P$ and $Q$ is given by the matrix $B$ as follows:

$B=\begin{array}{ccc}\mathrm{sms}& \mathrm{telephone}& \mathrm{letter}\\ 5000& 2500& 1000 \mathrm{P}\\ 8000& 1300& 2500 \mathrm{Q}\end{array}$

Find the total amount spent by the party in the two cities.