Let A and B be skew-symmetric matrices. This means that \(A^T = -A\) and \(B^T = -B\).

We want to determine if \(AB + BA\) is symmetric or skew-symmetric. To do this, we will find the transpose of \(AB + BA\).

\((AB + BA)^T = (AB)^T + (BA)^T\)

Using the property that \((XY)^T = Y^T X^T\), we have:

\((AB)^T = B^T A^T\) and \((BA)^T = A^T B^T\)

So, \((AB + BA)^T = B^T A^T + A^T B^T\)

Since A and B are skew-symmetric, \(A^T = -A\) and \(B^T = -B\). Substituting these into the equation:

\((AB + BA)^T = (-B)(-A) + (-A)(-B)\)

\((AB + BA)^T = BA + AB\)

\((AB + BA)^T = AB + BA\)

Since \((AB + BA)^T = AB + BA\), the matrix \(AB + BA\) is symmetric.