The sum of the eigen values of the matrix given below is \[\left[ {\begin{array}{*{20}{c}} 1&2&3 \\ 1&5&1 \\ 3&1&1 \end{array}} \right].\] A. 5 B. 7 C. 9 D. 18

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 the elements on the main diagonal. In this case, the matrix is $\left[ {\begin{array}{*{20}{c}} 1&2&3 \ 1&5&1 \ 3&1&1 \end{array}} \right]$. The trace of this matrix is $1+5+1=7$. Therefore, the sum of the eigenvalues of this matrix is $\boxed{7}$.

Here is a brief explanation of each option:

• Option A: $5$. This is the sum of the first and third elements on the main diagonal. However, the sum of the eigenvalues of a matrix is equal to the trace of the matrix, which is the sum of the elements on the main diagonal. In this case, the trace of the matrix is $1+5+1=7$. Therefore, the sum of the eigenvalues of this matrix is not $5$.
• Option B: $7$. This is the trace of the matrix. Therefore, the sum of the eigenvalues of this matrix is $7$.
• Option C: $9$. This is the sum of the first, second, and third elements on the main diagonal. However, the sum of the eigenvalues of a matrix is equal to the trace of the matrix, which is the sum of the elements on the main diagonal. In this case, the trace of the matrix is $1+5+1=7$. Therefore, the sum of the eigenvalues of this matrix is not $9$.
• Option D: $18$. This is the product of the first, second, and third elements on the main diagonal. However, the sum of the eigenvalues of a matrix is not equal to the product of the elements on the main diagonal. In general, the sum of the eigenvalues of a matrix is equal to the trace of the matrix, which is the sum of the elements on the main diagonal.