Q.Find the trace of the matrix $A = \begin{bmatrix} 1 & 3 & -5 \\ 2 & -1 & 5 \\ 2 & 0 & 1 \end{bmatrix}$.

Trace of a Matrix

The trace of a square matrix $A=[a_{ij}]$ of order $n$ is the sum of its main-diagonal

entries: $\operatorname{Tr}(A)=\sum_{i=1}^{n} a_{ii}$. It is a linear operation:

$\operatorname{Tr}(A+B)=\operatorname{Tr}(A)+\operatorname{Tr}(B)$ and

$\operatorname{Tr}(kA)=k\operatorname{Tr}(A)$. A useful non-obvious property is the

cyclic invariance $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$, from which

$\operatorname{Tr}(P^{-1}AP)=\operatorname{Tr}(A)$ — the trace is unchanged by

similarity and equals the sum of the eigenvalues.

To evaluate a trace when only combinations of matrices are given (say $B+C$ and

$B-C$), first recover the matrices, $B=\tfrac12(P+Q)$, $C=\tfrac12(P-Q)$, then compute

the required product's diagonal only — there is no need for the whole product matrix.

Elementary row operations that merely reorder or combine rows generally change the …