Show that if \(A\) and \(B\) are symmetric \(n \times n\) matrices, then so are \(A + B\) and \(A - B\).

Let \(C = A + B\) and \(D = A - B\). \(A\) and \(B\) being symmetric implies that \(a_{ij} = a_{ji}\) and \(b_{ij} = b_{ji}\). So for matrix \(C\) we have:

\[\begin{split} c_{ij} &= a_{ij} + b_{ij} \\ &= a_{ji} + b_{ji} \\ &= c_{ji} \\ \end{split}\]

and for matrix \(D\) we have:

\[\begin{split} d_{ij} &= a_{ij} - b_{ij} \\ &= a_{ji} - b_{ji} \\ &= d_{ji} \\ \end{split}\]

Therefore \(A+B\) and \(A-B\) are symmetric matrices.