1. **Problem:** A symmetric matrix $A$ has eigenvalues given. Given the eigenvectors corresponding to eigenvalues, determine: i. The eigenvector corresponding to a specific eigenvalue. ii. The matrix $A$. **Solution:** 1. Identify the eigenvalues and their corresponding eigenvectors. 2. The eigenvector corresponding to each eigenvalue is provided or found by solving $(A - \lambda I)x=0$. 3. For a symmetric matrix $A$, spectral theorem states $A = Q\Lambda Q^T$ where $Q$ contains normalized eigenvectors and $\Lambda$ is diagonal with eigenvalues. 4. Construct $Q$ matrix from eigenvectors and $\Lambda$ from eigenvalues. 5. Compute $A = Q \Lambda Q^T$. 2. **Problem:** Given $x_n$ is an approximation to the root of the equation $f(x) = 0$: i. Use Newton-Raphson method to show a better approximation is given by $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ ii. By taking $x_0=$ initial approximation, determine the root correct to 4 decimal places. **Solution:** 1. The Newton-Raphson method formula is derived from the tangent line approximation. 2. Successive approximations are calculated by the formula above. 3. Substitute $x_0$ into the formula to get $x_1$, then continue until difference between $x_{n+1}$ and $x_n$ is less than $10^{-4}$. 3. **Problem:** Using Table 1 representing a cubic polynomial $f(x)$ with tabulated $x$ and $f(x)$ values, use Newton-Gregory forward difference interpolation formula to find $f(x)$. **Solution:** 1. Calculate the forward differences $\Delta f(x)$, $\Delta^2 f(x)$, etc from the table. 2. Use the formula: $$ f(x) = f(x_0) + p \Delta f(x_0) + \frac{p(p-1)}{2!} \Delta^2 f(x_0) + \frac{p(p-1)(p-2)}{3!} \Delta^3 f(x_0) + \ldots $$ where $p = \frac{x - x_0}{h}$ and $h$ is the step size. 3. Substitute and simplify to get interpolation polynomial. 4. **Problem:** A linear time-variant system is described by differential equations: i. Express the system in matrix form $\dot{x} = A(t)x + B(t)u$. ii. Determine the state transition matrix $\Phi(t,t_0)$. **Solution:** 1. Write the system of differential equations in matrix form identifying $A(t)$ and $B(t)$. 2. The state transition matrix satisfies: $$ \frac{d}{dt} \Phi(t, t_0) = A(t) \Phi(t, t_0), \quad \Phi(t_0, t_0) = I $$ 3. Solve this differential matrix equation to find $\Phi(t,t_0)$. 5. **Problem:** Given functions $u$ and $v$: i. Express them in complex form $f(z) = u + iv$. ii. Show that $u$ and $v$ satisfy the Cauchy-Riemann equations. **Solution:** 1. Form complex function $f(z)$ using given $u$ and $v$. 2. Compute partial derivatives: $$ u_x, u_y, v_x, v_y $$ 3. Check Cauchy-Riemann equations: $$ u_x = v_y, \quad u_y = -v_x $$ 4. If satisfied, $f(z)$ is analytic. Final Answers: 1. $A = Q \Lambda Q^T$ with $Q$, $\Lambda$ from eigenvectors/eigenvalues. 2. $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ root approx. to 4 decimals. 3. Newton-Gregory polynomial from forward differences. 4. State equation and $\Phi(t,t_0)$ solution. 5. Verified Cauchy-Riemann for $u,v$.