Linear Algebra Diagonalize (find characteristic polynomial, eigenvalues, eigenvectors, ma- trix \( P \) and a diagonal matrix \( D \) such that \( A=P D P^{-1} \). Find \( P^{-1} \) and verify that the product \( P D P^{-1} \) is equal to \( A \) ) the following matrix \[ A=\left(\begin{array}{cccc}0 & 0 & 2 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1\end{array}\right) \]

75. Questions in which only two alternatives are possible is called A. Multiple choice questions B. Dichotomous questions C. Open ended questions D. Structured questions

Part-III: Choose the best answer from given choice (1pt eacil) 11. Let \( \mathrm{B}=\{\emptyset,\{\emptyset,\{\emptyset\}\}\} \). Then the number of proper subset is \( \begin{array}{llll}\text { A. } 8 & \text { B. } 7 & \text { C. } 3 & \text { D. } 4\end{array} \)

Explain the difference between exponential growth and exponential decay.

Use ajdoint matrix to evaluate the inverse of matrix \( A=\left(\begin{array}{cccc}3 & -3 & 2 & 4 \\ -4 & 4 & -2 & -2 \\ 2 & -3 & -3 & -3 \\ -1 & 2 & -4 & -3\end{array}\right) \)

[The most general such \( R \) is \( U \sqrt{A} \), where \( U \) is some isometry.] 6- Let \( x_{1}, \ldots, x_{n} \) be \( n \) vectors. Let \( G \) be a matrix with elements \( G_{i j}=\left(x_{i}, x_{j}\right) \). Show that \( G \) is a positive matrix. [ \( G \) is called the Gram matrix.]