Question

if the trace of a matrix is given as $tr(A)=2+i$, then $tr\left[ \left( 2-i \right)A \right]=$
A. $2+i$
B. $2-i$
C. $3$
D. $5$

Answer 1

Hint:- $tr(A)=2+i$ and we have to find $tr\left[ \left( 2-i \right)A \right]$. We know that the $tr(cA)$ is given by the formulae $tr(cA)=c\times tr(A)$. As we know the values of $c=2-i$ and $tr(A)=2+i$. So, by multiplying these two terms we will get the required $tr\left[ \left( 2-i \right)A \right]$.

Complete step-by-step solution -
Given, $tr(A)=2+i$
We have to find the $tr\left[ \left( 2-i \right)A \right]$
We know that the formulae for $tr(cA)$ is given by $tr(cA)=c\times tr(A)$. . . . . . .. . . (1)
So here in this problem $c=2-i$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . (2)
And $tr(A)=2+i$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(3)
$tr\left[ \left( 2-i \right)A \right]=$ $(2-i)(2+i)$ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . (4)
$={{2}^{2}}-{{i}^{2}}$
$=4+1$
$=5$
So the correct option is option(D)

Note: The trace of a square matrix is defined as the sum of the elements on the main diagonal. Some other properties of trace of a matrix is that the trace is called linear mapping means $tr(A+B)=tr(A)+tr(B)$and $tr(cA)=c\times tr(A)$.