Question

The sum of the eigen values of the square matrix $ \begin{pmatrix} {1} & {1} & {3}\\ {1} & {5} & {1}\\ {3} & {1} & {1} \end{pmatrix}$ (in integer).

Answer 1

Correct Answer: 7

To find the sum of the eigenvalues of a square matrix, we leverage an important property: the sum of the eigenvalues of a matrix is equal to the trace of the matrix. The trace of a matrix is the sum of its diagonal elements. Let's apply this to the given matrix \( A = \begin{pmatrix} 1 & 1 & 3\\ 1 & 5 & 1\\ 3 & 1 & 1 \end{pmatrix} \).

First, identify the diagonal elements of the matrix: \( a_{11}, a_{22}, a_{33} \), which are 1, 5, and 1, respectively.

Calculate the trace (sum of diagonal elements):
\( \text{Trace}(A) = a_{11} + a_{22} + a_{33} = 1 + 5 + 1 = 7 \).

Therefore, the sum of the eigenvalues of the matrix is 7.