Let A+2B=[ … ] and 2A-B=[ … ]. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A)-Tr(B) has value equal to

$$A+2B=[\begin{matrix}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{matrix}]\begin{matrix} \\ \\ \end{matrix}\ldots (1)$$A+2B=[16−5​2−33​031​]​…(1)

$$2A-B=[\begin{matrix}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{matrix}]\begin{matrix} \\ \\ \end{matrix}$$2A−B=[220​−1−11​562​]​

$$\Rightarrow 4A-2B=[\begin{matrix}4 & -2 & 10 \\ 4 & -2 & 12 \\ 0 & 2 & 4\end{matrix}]\ldots (2)$$⇒4A−2B=[440​−2−22​10124​]…(2)

$$(1)+(2)\Rightarrow 5A=[\begin{matrix}5 & 0 & 10 \\ 10 & -5 & 15 \\ -5 & 5 & 5\end{matrix}]$$(1)+(2)⇒5A=[510−5​0−55​10155​]

$A=[\begin{matrix}1 & 0 & 2 \\ 2 & -1 & 3 \\ -1 & 1 & 1\end{matrix}]$A=[12−1​0−11​231​] and $2A=[\begin{matrix}2 & 0 & 4 \\ 4 & -2 & 6 \\ -2 & 2 & 2\end{matrix}]$2A=[24−2​0−22​462​]

$$\therefore B=[\begin{matrix}2 & 0 & 4 \\ 4 & -2 & 6 \\ -2 & 2 & 2\end{matrix}]-[\begin{matrix}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{matrix}]$$∴B=[24−2​0−22​462​]−[220​−1−11​562​]

$$B=[\begin{matrix}0 & 1 & -1 \\ 2 & -1 & 0 \\ -2 & 1 & 0\end{matrix}]$$B=[02−2​1−11​−100​]

$$tr(A)=1-1+1=1$$tr(A)=1−1+1=1

$$tr(B)=-1$$tr(B)=−1

$tr(A)=1$tr(A)=1 and $tr(B)=-1$tr(B)=−1

$$\therefore tr(A)-tr(B)=2$$∴tr(A)−tr(B)=2