Chapter 7 - Eigenvalues and Eigenvectors - 7.3 Symmetric Matrices and Orthogonal Diagonalization - 7.3 Exercises - Page 370: 5

symmetric

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