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

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### 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

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#### 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 

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#### 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).

i) The eigenvalues of a Hermitian matrix are...
- Options:
  a) Purely real
  b) Zero
  c) Purely imaginary
  d) None of these

ii) If 2, 1, and λ are the eigenvalues of the matrix A = [a_{ij}] of order 3x3 and Trace(A) = 5, find the value of λ.
- Options:
  a) 4
  b) -1
  c) 0
  d) None of these

iii) Find the value of λ if the rank of matrix A = [[2, 2, 0], [4, 4, -2], [9, λ, 3]] is 3.

Explore solutions to problems involving Hermitian matrix eigenvalues, trace and determinant properties, and calculating matrix rank. Learn how to solve for unknown eigenvalues and maintain full matrix rank with detailed explanations.

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