Trace of a matrix and properties - Practice Questions & MCQ

Trace of the matrix:

The sum of all diagonal elements of a square matrix is called the trace of a matrix.
The trace of the matrix is denoted by $\operatorname{Tr}(\mathrm{A})$.

$
\operatorname{Tr}(\mathrm{A})=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{ii}}
$
For a given matrix A,

$
A=\left[\begin{array}{ccc}
-2 & 4 & 7 \\
8 & 3 & -1 \\
5 & -6 & 9
\end{array}\right], \quad \operatorname{Tr}(\mathrm{A})=-2+3+9=10
$
Properties of a trace of the matrix:
Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}} ; \mathrm{B}=\left[\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n} \text { and } \mathrm{k} \text { is a scalar, then }}$
i) $\operatorname{Tr}(\mathrm{kA})=\mathrm{k} \cdot \operatorname{Tr}(\mathrm{A})$
ii) $\operatorname{Tr}(\mathrm{A} \pm \mathrm{B})=\operatorname{Tr}(\mathrm{A}) \pm \operatorname{Tr}(\mathrm{B})$
iii) $\operatorname{Tr}(\mathrm{AB})=\operatorname{Tr}(B A)$
iv) $\operatorname{Tr}(\mathrm{A})=\operatorname{Tr}\left(\mathrm{A}^{\prime}\right)$
v) $\operatorname{Tr}(A B) \neq \operatorname{Tr}(A) \cdot \operatorname{Tr}(B)$