The image contains a set of questions related to matrices and eigenvalues. Let's address each part of the question step-by-step.

Question 1:

Part (i)

Question: The eigenvalues of a Hermitian matrix are...

• Options:
  • (a) Purely real
  • (b) Zero
  • (c) Purely imaginary
  • (d) None of these

Solution: For a Hermitian matrix, the eigenvalues are always real. This is a known property in linear algebra.

• Answer: (a) Purely real

Part (ii)

Question: If $2$, $1$, and $\lambda$ are the eigenvalues of the matrix $A = [a_{ij}]$ of order $3 \times 3$ and Trace($A$) = 5, find the value of $\lambda$.

• Options:
  • (a) 4
  • (b) -1
  • (c) 0
  • (d) None of these

Solution: The trace of a matrix (sum of its diagonal elements) is also equal to the sum of its eigenvalues. Given: $\text{Trace}(A) = 2 + 1 + \lambda = 5$ Solving for $\lambda$: $3 + \lambda = 5 \Rightarrow \lambda = 2$

• Answer: (a) 4

Part (iii)

Question: Find the value of $\lambda$ if the rank of matrix $A = \begin{bmatrix} 2 & 2 & 0 \\ 4 & 4 & -2 \\ 9 & \lambda & 3 \end{bmatrix}$ is 3.

Solution: To determine $\lambda$ such that the rank of $A$ is 3, the matrix must have full rank (no zero determinant of any principal submatrices).