Let A + 2B = [ cases 1 & 2 & 0 6 & - 3 & 3 - 5 & 3 & 1 cases ] and...

A + 2B = \left[\begin{cases}1 & 2 & 0 \\ 6 & - 3 & 3 \\ - 5 & 3 & 1\end{cases}\right] \ldots \left(1\right)

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

⇒ 4A - 2B = \left[\begin{cases}4 & - 2 & 10 \\ 4 & - 2 & 12 \\ 0 & 2 & 4\end{cases}\right] \ldots\left(2\right)$</p><p>$\left(1\right) + \left(2\right) ⇒ 5A = \left[\begin{cases}5 & 0 & 10 \\ 10 & - 5 & 15 \\ - 5 & 5 & 5\end{cases}\right]

A = \left[\begin{cases}1 & 0 & 2 \\ 2 & - 1 & 3 \\ - 1 & 1 & 1\end{cases}\right] and 2A = \left[\begin{cases}2 & 0 & 4 \\ 4 & - 2 & 6 \\ - 2 & 2 & 2\end{cases}\right]

∴ B = \left[\begin{cases}2 & 0 & 4 \\ 4 & - 2 & 6 \\ - 2 & 2 & 2\end{cases}\right] - \left[\begin{cases}2 & - 1 & 5 \\ 2 & - 1 & 6 \\ 0 & 1 & 2\end{cases}\right]

B = \left[\begin{cases}0 & 1 & - 1 \\ 2 & - 1 & 0 \\ - 2 & 1 & 0\end{cases}\right]

$tr ( A ) = 1 - 1 + 1 = 1$

$tr ( B ) = - 1$

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

$∴ tr ( A ) - tr ( B ) = 2$