Mathematics - Matrices Question with Solution | TestHub

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

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

$\Rightarrow 4A-2B=\left[\begin{array}{ccc}4& -2& 10\\ 4& -2& 12\\ 0& 2& 4\end{array}\right] \dots \left(2\right)$

$\left(1\right)+\left(2\right)\Rightarrow 5 \mathrm{A}=\left[\begin{array}{ccc}5& 0& 10\\ 10& -5& 15\\ -5& 5& 5\end{array}\right]$

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

$\therefore B=\left[\begin{array}{ccc}2& 0& 4\\ 4& -2& 6\\ -2& 2& 2\end{array}\right]-\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$

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

$\mathit{tr}\left(A\right)=1-1+1=1$

$tr\left(B\right)=-1$

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

$\therefore tr\left(A\right)-tr\left(B\right)=2$