Question

$A=\left|\begin{array}{cc}1& 1\\ 2& 2\\ 3& -1\end{array}\right|$ and $B=\left|\begin{array}{cc}2& 1\\ 1& 2\\ -1& 0\end{array}\right|$, then find matrix $C$ if $A+2B+C=0$.

Answer 1

$2B=2\left|\begin{array}{cc}2& 1\\ 1& 2\\ -1& 0\end{array}\right|=\left|\begin{array}{cc}4& 2\\ 2& 4\\ -2& 0\end{array}\right|$

$A+2B=\left|\begin{array}{cc}1& 1\\ 2& 2\\ 3& -1\end{array}\right|+\left|\begin{array}{cc}4& 2\\ 2& 4\\ -2& 0\end{array}\right|=\left|\begin{array}{cc}1+4& 2+1\\ 2+2& 4+2\\ 3+(-2)& (-1)+0\end{array}\right|=\left|\begin{array}{cc}5& 3\\ 4& 6\\ 1& -1\end{array}\right|$

$C=-(A+2B)=\left|\begin{array}{cc}5& 3\\ 4& 6\\ 1& -1\end{array}\right|\Rightarrow \left|\begin{array}{cc}-5& -3\\ -4& -6\\ -1& 1\end{array}\right|$

$\therefore A+2B+C=0$