# Trace of Sum of Matrices is Sum of Traces

## Theorem

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 $\mathbf B$.

Then:

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

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

## Proof

 $\ds \map \tr {\mathbf A} + \map \tr {\mathbf B}$ $=$ $\ds \sum_{k \mathop = 1}^n a_{kk} + \sum_{k \mathop = 1}^n b_{kk}$ 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

$\blacksquare$