## Question

Let P be a real square matrix satisfying and .

i) Can the matrix P have complex eigenvalues? If so, construct an example, else, justify your answer.

ii) What are the eigenvalues of P?

## Solution

### Part-1: Conceptualize the Matrix P

implies that P is a Symmetric Matrix. This means that except the diagonal elements, all other elements of the matrix are like .

Thus,

Let’s consider this sample. Assuming that whatever non-size related property that holds true for this 2 x 2 matrix should hold true for an n x n matrix as well.

### Part-2: Construct the Matrix P

We know that,

Thus,

(1)

(2)

(3)

(4)

(4) & (3)

Thus, matrix P is an Identity matrix.

### Part-3: Find Eigenvalues of P

*Thus, Eigenvalue of P is λ=1*

### Part-4: Can the matrix P have complex eigenvalues?

*From Part 3, we see that P cannot have complex values as eigenvalues.*