question_answer

The trace \[{{\text{T}}_{\text{r}}}\] of a \[3\times 3\] matrix A = \[\text{(}{{\text{a}}_{\text{ij}}}\text{)}\] is defined by the relation \[{{\text{T}}_{\text{r}}}\] \[\text{=}{{\text{a}}_{\text{11}}}\text{+}{{\text{a}}_{\text{22}}}\text{+}{{\text{a}}_{\text{33}}}\] (i.e. \[{{\text{T}}_{\text{r}}}\] is sum of the main diagonal elements). Which of the following statements cannot hold?

A)  \[{{\text{T}}_{\text{r}}}\text{(kA)=k}{{\text{T}}_{\text{r}}}\text{(A)}\](k is a scalar)

B)  \[{{\text{T}}_{\text{r}}}\text{(A+B)}\,\text{=}\,{{\text{T}}_{\text{r}}}\text{(A)+}{{\text{T}}_{\text{r}}}(\text{B)}\]

C)  \[{{\text{T}}_{\text{r}}}\text{(}{{\text{I}}_{\text{3}}}\text{)}\,\text{=3}\]          

D)  \[{{\text{T}}_{\text{r}}}\text{(}{{\text{A}}^{\text{2}}}\text{)}\,\text{=}{{\text{T}}_{\text{r}}}{{\text{(A)}}^{\text{2}}}\]