How do you calculate the trace of a matrix?

The trace of a matrix is defined as the sum of the elements on its main diagonal, which runs from the top left corner to the bottom right corner of the matrix. This means that for a square matrix ( A ), the trace is calculated as:

[ \text{Trace}(A) = a_{11} + a_{22} + a_{33} + \ldots + a_{nn}

]

where ( a_{ii} ) represents the elements along the diagonal of the matrix. Thus, the correct method to compute the trace is simply to add these diagonal elements together.

Choosing to multiply all elements along the diagonal instead does not yield the correct quantity representing the trace. The operations of averaging or taking the square root of sums would also not relate to how the trace is defined or calculated. The trace provides important information in linear algebra, especially regarding properties of matrices like eigenvalues, but it is distinctly about summation and not multiplication or averaging.