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

• A. $0$
• B. $1$
• C. $2$
• D. none

Answer 1

The correct option is C $2$

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

Let $\text{tr}\left(A\right)=x$ and $\text{tr}\left(B\right)=y.$ Then,
$x+2y=-1$ and $2x-y=3$
On solving, we get
$x=1$ and $y=-1.$

Hence, $\text{tr}\left(A\right)-\text{tr}\left(B\right)=x-y=2$