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 \[tr(cA)=c\times tr(A)\].