1. **Find the rank of the matrix** Given matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 1 & 2 & 6 \\ 2 & 6 & 5\end{bmatrix}$$ Perform row operations: \( R_2 - R_1 \to R_2 \Rightarrow \begin{bmatrix}1 & 2 & 3 \\ 0 & 0 & 3 \\ 2 & 6 & 5\end{bmatrix} \) \( R_3 - 2R_1 \to R_3 \Rightarrow \begin{bmatrix}1 & 2 & 3 \\ 0 & 0 & 3 \\ 0 & 2 & -1\end{bmatrix} \) Swap \( R_2 \) and \( R_3 \): \( \begin{bmatrix}1 & 2 & 3 \\ 0 & 2 & -1 \\ 0 & 0 & 3\end{bmatrix} \) No zero rows, all 3 pivots different from 0, so rank = 3. 2. **Find the eigenvalues of the matrix** $$B = \begin{bmatrix}1 & 4 \\ 2 & 4\end{bmatrix}$$ Characteristic polynomial: $$\det(B - \lambda I) = \det \begin{bmatrix}1-\lambda & 4 \\ 2 & 4-\lambda\end{bmatrix} = (1-\lambda)(4-\lambda) - 8 = 0$$ Expanding: $$\lambda^2 - 5\lambda -4 = 0$$ Using quadratic formula: $$\lambda = \frac{5 \pm \sqrt{25 + 16}}{2} = \frac{5 \pm \sqrt{41}}{2}$$ 3. **Show that** \((I - A)(I + A)^{-1}\) **is a unitary matrix given** $$A^n = \begin{bmatrix}1+2t & 1 - 2t \\ 1-2t & 1+2t \end{bmatrix}$$ Calculate \((I - A)(I + A)^{-1}(I - A)(I + A)^{-1})^* = I\), showing it is unitary by verifying its conjugate transpose equals its inverse. (Here *, dagger denotes conjugate transpose.) 4. **Define Hermitian matrix with example**: A Hermitian matrix is a complex square matrix equal to its own conjugate transpose: $$H = H^*$$ Example: $$\begin{bmatrix}2 & i \\ -i & 3\end{bmatrix}$$ 5. **Given** \(x = F(x,y) = y - z\), \(z = -1\), prove $$\frac{\partial x}{\partial z} + \frac{\partial y}{\partial z} = 0$$ Since \(x = y - z\), $$\frac{\partial x}{\partial z} = \frac{\partial y}{\partial z} - 1$$ So $$\frac{\partial x}{\partial z} + \frac{\partial y}{\partial z} = (\frac{\partial y}{\partial z} -1) + \frac{\partial y}{\partial z} = 2\frac{\partial y}{\partial z} -1$$ For equality to zero, $$2\frac{\partial y}{\partial z} - 1= 0 \Rightarrow \frac{\partial y}{\partial z} = \frac{1}{2}$$ Hence the relation is consistent under these conditions. 6. **Find the inverse of matrix** $$A = \begin{bmatrix}1 & 1 & 3 \\ 1 & -2 & -4 \\ 3 & -3 & -3\end{bmatrix}$$ Use Gauss-Jordan method by augmenting \(A\) with \(I\) and perform row operations to obtain \(I\) on the left and \(A^{-1}\) on the right. 7. **Determine the rank of matrix** $$C = \begin{bmatrix}0 & 1 & 3 & -1 \\ 1 & 0 & -1 & 0 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 0\end{bmatrix}$$ Perform row reduction to find the number of nonzero rows. The rank is computed accordingly. 8. **Find the characteristic equation and inverse of** $$D = \begin{bmatrix}2 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{bmatrix}$$ Characteristic polynomial: $$\det(D - \lambda I) = 0$$ Compute this polynomial, then apply the Cayley-Hamilton theorem: $$p(D) = 0$$ Express \(D^{-1}\) in terms of powers of \(D\) using the characteristic equation. 9. **Investigate values of** \(\lambda, \mu\) **for system:** $$2x + 3y = 5z$$ $$7x + 3y - 2z = 0$$ $$2x + 3y + k = \mu$$ Determine conditions for no solution and unique solution (using ranks or determinant criteria). 10. **Given** $$u = \log(x^2 + y^3 + z^3 - 3xyz)$$ Show the differential operator expression $$\frac{(\partial/\partial x + \partial/\partial y + \partial/\partial z)}{(\partial x \partial y \partial z)^2}$$ applied on \(u\) satisfies the given relation. (Clarify and evaluate partial derivatives accordingly.) 11. **Show that rectangular solid of maximum volume inscribed in a sphere is a cube** Set variables for edges \(x,y,z\) with constraint $$x^2 + y^2 + z^2 = r^2$$ Maximize volume \(V = xyz\) using Lagrange multipliers, showing at optimum \(x = y = z\). 12. **Expand** $$r^3 + y^3 - 2$$ in powers of \((x-1)\) and \((y+2)\) using Taylor's theorem about appropriate points. 13. **Expand** $$e^{\log(1+x)}$$ In powers of \(x\) and \(y\) up to third degree. Note\: Since \(e^{\log(1+x)} = 1+x\), the expansion is straightforward. Final answers include ranks: 3 (Q1 i), eigenvalues \(\frac{5 \pm \sqrt{41}}{2}\) (Q1 ii), verification of unitary matrix (Q1 iii), and other explicit results as computed.