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 $tr\left(A\right)-tr\left(B\right)$ has the value equal to

• A. 0
• B. 1
• C. 2
• D. none of these

Answer 1

The correct option is C 2

$A+2B=\left[\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$ .....(i)
$2A-B=\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$ .....(ii)
$\Rightarrow 4A-2B=\left[\begin{array}{ccc}4& -2& 10\\ 4& -2& 12\\ 0& 2& 4\end{array}\right]$ ....(iii)
Adding (i) and (iii), we get
$5A=\left[\begin{array}{ccc}5& 0& 10\\ 10& -5& 15\\ -5& 5& 5\end{array}\right]$
$\Rightarrow A=\left[\begin{array}{ccc}1& 0& 2\\ 2& -1& 3\\ -1& 1& 1\end{array}\right]$
So, $tr\left(A\right)=1$
Now, by eq(ii),
$B=\left[\begin{array}{ccc}2& 0& 4\\ 4& -2& 6\\ -2& 2& 2\end{array}\right]-\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$
$\Rightarrow B=\left[\begin{array}{ccc}0& 1& -1\\ 2& -1& 0\\ -2& 1& 0\end{array}\right]$
So, $tr\left(B\right)=-1$
Now, $tr\left(A\right)-tr\left(B\right)=1+1=2$