1. Example Using Gauss-Jordan method determine the inverse of the ma- trix \[ A=\left[\begin{array}{ccc}5 & 2 & -3 \\ 3 & 1 & -2 \\ 2 & -3 & -4\end{array}\right] \in \mathbb{R}^{3 \times 3} \]

\( \mathrm{U}=\{1,2, \mathbf{3}, \ldots \mathbf{1 0}\} \) 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.

1ple 3: Prove that \( (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \) (i) Using Venn diagrams (ii) rigorously

c) \( \begin{array}{l}\mathrm{U}=\{1,2,3, \ldots 10\} \text { is a given universal set. } O=\{1,3,5,7,9\} \text { and } \\ \mathrm{P}=\{\mathbf{3}, \mathbf{5}, 7\} \text { are any two given sets. } \\ \text { (i) Is the set } O \text { a subset of } U \text { ? Give reason. } \\ \text { (ii) Is the set } P \text { a proper or improper subset of } O \text { ? Give reason. } \\ \text { (iii) Show the sets } O \text { and } P \text { in a diagram. }\end{array} \).

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)

10. Найти собственные значения и собственные и присоединенные векторы матрицы лнейного оператора и найти вид этой матрицы в базисе пз собственных и присоединенных векторов \[ \left[\begin{array}{rrr}5 & 4 & -4 \\ -1 & 3 & 1 \\ 0 & 2 & 2\end{array}\right] \]