## Questions About the Trace of a Matrix

## Problem 19

Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.

**(a)** Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$.

**(b)** Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.

**(c)** Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.