If $A = \left[ {\begin{array}{*{20}{c}}
\alpha &0\\
1&1
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
1&0\\
5&1
\end{array}} \right]$ , then the value of $\alpha $ for which $A^2 = B$ is

If $A$=$\left[ {\begin{array}{*{20}{c}}2&0&0\\0&2&0\\0&0&2\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}1&2&3\\0&1&3\\0&0&2\end{array}} \right],$ then $|AB|$ is equal to

If $\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right],$ then prove that $\mathrm{A}^{n}=\left[\begin{array}{cc}\cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta\end{array}\right], n \in \mathrm{N}$

A trust fund has Rs. $30,000$ that must be invested in two different types of bonds. The first bond pays $5 \%$ interest per year, and the second bond pays $7 \%$ interest per year. Using matrix multiplication, determine how to divide Rs. $30,000$ among the two types of bonds. If the trust fund must obtain an annual total interest of  Rs.  $1800$.

If $\omega \ne 1$ is cube root of unity and $H = \left[ {\begin{array}{*{20}{c}}\omega &0\\0&\omega \end{array}} \right]$ then ${H^{70}}$ is equal to