91Ó°ÊÓ

Let \(A\) and \(B\) be symmetric \(n \times n\) matrices. Prove that \(A B=B A\) if and only if \(A B\) is also symmetric.

Let \(A\) be an \(n \times n\) matrix and let $$B=A+A^{T} \quad \text { and } \quad C=A-A^{T}$$ (a) Show that \(B\) is symmetric and \(C\) is skew symmetric. (b) Show that every \(n \times n\) matrix can be represented as a sum of a symmetric matrix and a skew-symmetric matrix.

Consider the matrix $$A=\left(\begin{array}{lllll} 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 \end{array}\right)$$ (a) Draw a graph that has \(A\) as its adjacency matrix. Be sure to label the vertices of the graph. (b) By inspecting the graph, determine the number of walks of length 2 from \(V_{2}\) to \(V_{3}\) and from \(V_{2}\) to \(V_{5}\) (c) Compute the second row of \(A^{3}\) and use it to determine the number of walks of length 3 from \(V_{2}\) to \(V_{3}\) and from \(V_{2}\) to \(V_{5}\)

Consider the graph (a) Determine the adjacency matrix \(A\) of the graph. (b) Compute \(A^{2}\). What do the entries in the first row of \(A^{2}\) tell you about walks of length 2 that start from \(V_{1} ?\) (c) Compute \(A^{3}\). How many walks of length 3 are there from \(V_{2}\) to \(V_{4}\) ? How many walks of length less than or equal to 3 are there from \(V_{2}\) to \(V_{4} ?\)