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\)
Section 3.8 Trace
The trace of a (square) matrix is just the sum of all of its diagonal elements. In terms of components,
\begin{equation}
\mathrm{tr}(A)=\sum_i A_{ii}\text{.}\tag{3.8.1}
\end{equation}
For example, if
\begin{equation}
A=\begin{pmatrix}
1\amp 2\amp 3\\
4\amp 5\amp 6\\
7\amp 8\amp 9
\end{pmatrix}\tag{3.8.2}
\end{equation}
then
\begin{equation}
\mathrm{tr}(A)=1+5+9=15\text{.}\tag{3.8.3}
\end{equation}
Checkpoint 3.5. Try it yourself: Trace.
Compute \(\mathrm{tr}(B)\) if
\begin{equation}
B = \begin{pmatrix}
1\amp 34\amp 5\\
23\amp 5\amp 98\\
132\amp 7\amp 9
\end{pmatrix}\text{.}\tag{3.8.4}
\end{equation}
Solution.
\begin{equation}
\tr(B) = 1 + 5 + 9 = 15\text{.}\tag{3.8.5}
\end{equation}
