Linear Algebra

Step 1: Use determinant condition.

Step 2: Write the matrix.

Step 3: Compute eigenvalues.
Solve :
Eigenvalues are and .

Step 4: Conclusion.
One eigenvalue is 13 ⇒ option (C).

Step 1: Understanding eigenvalues.
To find the eigenvalues of the matrix, we solve the characteristic equation:

where is the matrix, is the eigenvalue, and is the identity matrix.
Step 2: Solving for eigenvalues.
For the given matrix:

we find the characteristic equation by subtracting from the diagonal elements and computing the determinant. The resulting eigenvalues are .
Step 3: Conclusion.
Thus, the eigenvalues of the matrix are and , so the correct answer is (C).

Step 1: Recall the property of Hermitian matrices.
A matrix is Hermitian if . For two Hermitian matrices and , we have and .

Step 2: Check .
Thus, is Hermitian.

Step 3: Check .
Hence, is Hermitian.

Step 4: Check the remaining options.
- is not necessarily Hermitian because . - is anti-Hermitian since .

Step 5: Conclusion.
Hence, the correct statements are (A) and (B).

Step 1: Understanding the Concept:
A system of homogeneous linear equations (where all constant terms are zero) of the form has a non-trivial solution (a solution other than ) if and only if the determinant of the coefficient matrix is zero.
Step 2: Key Formula or Approach:
We need to set up the coefficient matrix for the given system of equations and then calculate its determinant. The condition for a non-trivial solution is .
Step 3: Detailed Explanation:
The given system of equations is: The coefficient matrix is: For a non-trivial solution, the determinant of must be zero. Let's compute the determinant by expanding along the first row: Combine like terms: Since it is given that is a non-zero constant, we can divide the entire equation by : Rearranging the terms to match the form of the options: This condition must be satisfied for the system to have a non-trivial solution. Step 4: Final Answer:
The condition for a non-trivial solution is . This corresponds to option (B).