Trace of Unit Matrix

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Theorem

Let $\mathbf I_n$ be the unit matrix of order $n$.


Then:

$\operatorname{tr} \left({\mathbf I_n}\right) = n$

where $\operatorname{tr} \left({\mathbf I_n}\right)$ denotes the trace of $\mathbf I_n$.


Proof

By definition:

$\mathbf I_n := \left[{a}\right]_n: a_{i j} = \delta_{i j}$

That is: each of the elements on the main diagonal is equal to $1$.

There are $n$ such elements.

Hence the result.

$\blacksquare$


Sources