Question

Given $A=\left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right],B=\left[\begin{array}{cc}-3& 2\\ 4& 0\end{array}\right]andC=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right],$ find the matrix $X$ such that $A+X=2B+C.$

• A. $\left[\begin{array}{cc}5& -7\\ 2& 6\end{array}\right]$
• B. $\left[\begin{array}{cc}-7& 2\\ 6& 5\end{array}\right]$
• C. $\left[\begin{array}{cc}-6& 5\\ 6& 2\end{array}\right]$
• D. $\left[\begin{array}{cc}-7& 6\\ 5& 2\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{cc}-6& 5\\ 6& 2\end{array}\right]$

$A+X=2B+C$

Let $X=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]+\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=2\left[\begin{array}{cc}-3& 2\\ 4& 0\end{array}\right]+\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}2+a& -1+b\\ 2+c& d\end{array}\right]=\left[\begin{array}{cc}-4& 4\\ 8& 2\end{array}\right]$

$\Rightarrow 2+a=-4,-1+b=4$

$\Rightarrow a=-6,b=5$

$\Rightarrow 2+c=8,d=2$

$\Rightarrow c=6$

$\therefore X=\left[\begin{array}{cc}-6& 5\\ 6& 2\end{array}\right]$

Hence, the answer is $\left[\begin{array}{cc}-6& 5\\ 6& 2\end{array}\right].$