Question

Let

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

and

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

If $Tr\left(A\right)$ denotes the sum of all diagonal elements of the matrix $A$, then $Tr\left(A\right)-Tr\left(B\right)$ has value equal to:

• A. $0$
• B. $1$
• C. $3$
• D. $2$

Answer 1

The correct option is D

$2$

Explanation for the correct option:

Find the value of the given expression.

In the question, two equations are given.

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

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

It is also given that, $Tr\left(A\right)$ denotes the sum of all diagonal elements of the matrix $A$.

Therefore, from equation $1$.

$$\begin{gathered} tr\left(A+2B\right)=-1 \\ \Rightarrow tr\left(A\right)+tr\left(2B\right)=-1 \\ \Rightarrow tr\left(A\right)+2tr\left(B\right)=-1...3 \end{gathered}$$

From equation $2$.

$$\begin{gathered} tr(2A-B)=3 \\ \Rightarrow tr\left(2A\right)-tr\left(B\right)=3 \\ \Rightarrow 2tr\left(A\right)-tr\left(B\right)=3...4 \end{gathered}$$

On solving equation $3$ and $4$ simultaneously.

We get $tr\left(A\right)=1$ and $tr\left(B\right)=-1$.

So the value of $Tr\left(A\right)-Tr\left(B\right)$ is $1-(-1)=2$.

Therefore, the value of the given expression is $2$.

Hence, option $D$ is the correct answer.