Question

Given $A=\left[\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}3& -1& 2\\ 4& 2& 5\\ 2& 0& 3\end{array}\right]$.
Find the matrix $C$ such that $A+2C=B$
If $C=\left[\begin{array}{ccc}1& \frac{-3}{2}& a\\ \frac{-1}{2}& 1& b\\ \frac{1}{2}& \frac{1}{2}& 1\end{array}\right]$
Find the value of a+b?

Answer 1

Given that $A+2C=B$

$\therefore 2C=B-A$

$=\left[\begin{array}{ccc}3& -1& 2\\ 4& 2& 5\\ 2& 0& 3\end{array}\right]-\left[\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array}\right]=\left[\begin{array}{ccc}3-1& -1-3& 2+3\\ 4-5& 2-0& 5-2\\ 2-1& 0+1& 3-1\end{array}\right]$
$\therefore 2C=\left[\begin{array}{ccc}2& -3& 5\\ -1& 2& 3\\ 1& 1& 2\end{array}\right]$

$\Rightarrow C=\frac{1}{2}\left[\begin{array}{ccc}2& -3& 5\\ -1& 2& 3\\ 1& 1& 2\end{array}\right]=\left[\begin{array}{ccc}1& \frac{-3}{2}& \frac{5}{2}\\ \frac{-1}{2}& 1& \frac{3}{2}\\ \frac{1}{2}& \frac{1}{2}& 1\end{array}\right]$

$\Rightarrow a+b=4$