Trace of Sum of Matrices is Sum of Traces

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Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be square matrices of order $n$.

let $\mathbf A + \mathbf B$ debote the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.


$\map \tr {\mathbf A + \mathbf B} = \map \tr {\mathbf A} + \map \tr {\mathbf B}$

where $\map \tr {\mathbf A}$ denotes the trace of $\mathbf A$.


\(\ds \map \tr {\mathbf A} + \map \tr {\mathbf B}\) \(=\) \(\ds \sum_{k \mathop = 1}^n a_{kk} + \sum_{k \mathop = 1}^n b_{kk}\) Definition of Trace of Matrix
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \paren {a_{kk} + b_{kk} }\) Sum of Summations equals Summation of Sum
\(\ds \) \(=\) \(\ds \map \tr {\mathbf A + \mathbf B}\) Definition of Matrix Entrywise Addition, Definition of Trace of Matrix