Question

If $A=\text{diag}(2-59),B=\text{diag}(11-4)$ and $C=\text{diag}(-634)$, then find $B+C-2A$.

Answer 1

Here, $A=\left[\begin{array}{ccc}2& 0& 0\\ 0& -5& 0\\ 0& 0& 9\end{array}\right],$ $B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -4\end{array}\right]$ and $C=\left[\begin{array}{ccc}-6& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]$

$\Rightarrow$ $B+C-2A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -4\end{array}\right]+\left[\begin{array}{ccc}-6& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right]-2\left[\begin{array}{ccc}2& 0& 0\\ 0& -5& 0\\ 0& 0& 9\end{array}\right]$

$=\left[\begin{array}{ccc}1+(-6)& 0+0& 0+0\\ 0+0& 1+3& 0+0\\ 0+0& 0+0& -4+4\end{array}\right]-\left[\begin{array}{ccc}2\times 2& 0\times 2& 0\times 2\\ 0\times 2& -5\times 2& 0\times 2\\ 0\times 2& 0\times 2& 9\times 2\end{array}\right]$

$=\left[\begin{array}{ccc}-5& 0& 0\\ 0& 4& 0\\ 0& 0& 0\end{array}\right]-\left[\begin{array}{ccc}4& 0& 0\\ 0& -10& 0\\ 0& 0& 18\end{array}\right]$

$=\left[\begin{array}{ccc}-5-4& 0-0& 0-0\\ 0-0& 4-(-10)& 0-0\\ 0-0& 0-0& 0-18\end{array}\right]$

$=\left[\begin{array}{ccc}-9& 0& 0\\ 0& 14& 0\\ 0& 0& -18\end{array}\right]=diag(-914-18)$