If  A+2B=1206−33−531, and 2A−B=2−152−16012, then tr(A)−tr(B) is

# If  $A+2B=\left[\begin{array}{lll}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$, and $2A-B=\left[\begin{array}{lll}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$, then $\text{tr}\left(A\right)-\text{tr}\left(B\right)$ is

1. A

1

2. B

2

3. C

3

4. D

4

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### Solution:

$A+2B=\left[\begin{array}{lll}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$

$\text{tr}\left(A+2B\right)=-1$

$\text{tr}\left(A\right)+2\text{tr}\left(B\right)=-1$

$2A-B=\left[\begin{array}{lll}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$

$\text{tr}\left(2A-B\right)=3$

$2\text{tr}\left(A\right)-\text{tr}\left(B\right)=3$

Solving (1) and (2), we get  $\text{tr}\left(A\right)=1,\text{tr}\left(B\right)=-1$

$\text{tr}\left(A\right)-\text{tr}\left(B\right)=2$

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