1-) \( \left(20\right. \) p) Let \( A=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 3 & 4 \\ 5 & 2 & 0\end{array}\right] \) be a matrix. Find \( \operatorname{det}(A) \), adj \( A \) and the inverse of the matrix if it exists.

\( \mathrm{U}=\{1,2, \mathbf{3}, \ldots \mathbf{1 0}\} \) is a given universal set. \( \mathrm{O}=\{\mathbf{1}, \mathbf{3}, \mathbf{5}, \mathbf{7} \), \( \mathrm{P}=\{\mathbf{3}, \mathbf{5}, 7\} \) are any two given sets. (i) Is the set \( O \) a subset of \( U \) ? Give reason.

IX. Please determine whether the set of vectors \( \left\{\begin{array}{lll}\alpha_{1} & \alpha_{2} & \alpha_{3}\end{array}\right\} \) is linearly dependent or not, where \( \alpha_{1}=\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)^{T}, \alpha_{2}=\left(\begin{array}{llll}2 & 1 & 1\end{array}\right)^{T}, \alpha_{3}=\left(\begin{array}{lll}3 & 2 & 1\end{array}\right)^{T} \). (Write the answer for the questions on your answer sheet. 10 points)

Q. 15 Find the eigenvalues and eigenvectors of the matrix \( \left[\begin{array}{ll}6 & 5 \\ 3 & 4\end{array}\right] \).

4. If \( \quad\left[\begin{array}{cccccc}0 & 0 & -3 & -2 & 0 & 5 \\ -2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -2 & 3 & 0 & 0 & 0 \\ -2 & 8 & 3 & 6 & 2 & 7 \\ 0 & 2 & 3 & 0 & 0 & 0\end{array}\right] \operatorname{compute} \operatorname{det}(A),(\operatorname{adj}(A))^{-1} \) and \( \operatorname{det}(\operatorname{adj}(A)) \)

Decision variables in an OR model are : (a) Controllable (b) Uncontrollable (c) Parameters (d) Constants