Question

Let $A+2B=\left[\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$ and $2A-B=\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$, then find tr(A) - tr(B).

Answer 1

Given $A+2B=\left[\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$
$\Rightarrow tr(A+2B)=-1$
or $tr\left(A\right)+2tr\left(B\right)=-1$ .............. (1)
Also given $2A-B=\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$
$\Rightarrow tr(2A-B)=3$
or $2tr\left(A\right)-tr\left(B\right)=3$ ............ (2)
Solving (1) and (2), we get
$tr\left(A\right)=1$ and $tr\left(B\right)=-1$
$\Rightarrow tr\left(A\right)-tr\left(B\right)=2$