$$\text { If }\left|\begin{array}{lll} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{array}\right|=0 \text {, then the value of } \alpha \text { is: }$$
Case Study II
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop. Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250 .
Based on the above information, answer the following questions:
(i) Convert the given above situation into a matrix equation of the form $$A X=B$$.
(ii) Find $$|\mathrm{A}|$$.
(iii) Find $$\mathrm{A}^{-1}$$.
OR
(iii) Determine $$P=A^2-5 A$$.