Chapter 7 - Eigenvalues and Eigenvectors - 7.3 Symmetric Matrices and Orthogonal Diagonalization - 7.3 Exercises - Page 370: 6

symmetric

A matrix is symmetric if it is equal to its transpose. Here the transpose of the matrix is $\begin{bmatrix} 2& 0& 3&5 \\ 0& 11&0&-2\\ 3&0&5&0\\ 5&-2&0&1\\ \end{bmatrix} $, which is the same as the original matrix; thus it is symmetric.