# Find a matrix X such that 2A + 2B + X = 0, where $$A = \begin{bmatrix} -1 & 3 \\ 2 & 1 \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 5 \\ 3 & 2 \end{bmatrix}$$

Toolbox:
• The sum / difference $A(+/-)B$ of two $m$-by-$n$ matrices $A$ and $B$ is calculated entrywise: $(A (+/-) B)_{i,j} = A_{i,j} +/- B_{i,j}$ where $1 \leq i \leq m$ and $1 \leq j \leq n$
Step1:
Given:
$A=\begin{bmatrix}-1 & 3\\2 & 1\end{bmatrix}$ $B=\begin{bmatrix}1 & 5\\3 & 2\end{bmatrix}$
2A+2B+X=0
Let X=$\begin{bmatrix}a & b\\c & d\end{bmatrix}$
Step2:
$2\begin{bmatrix}-1 & 3\\2 & 1\end{bmatrix}+2\begin{bmatrix}1 & 5\\3 &2\end{bmatrix}+\begin{bmatrix}a & b\\c & d\end{bmatrix}=0$
$\begin{bmatrix}-2 & 6\\4 & 2\end{bmatrix}+\begin{bmatrix}2 & 10\\6 &4\end{bmatrix}+\begin{bmatrix}a & b\\c & d\end{bmatrix}=0$
$\begin{bmatrix}0 & 16\\10 & 6\end{bmatrix}+\begin{bmatrix}a & b\\c& d\end{bmatrix}=0$
$\begin{bmatrix}a & b\\c& d\end{bmatrix}=\begin{bmatrix}0 & -16\\-10 & -6\end{bmatrix}$
$X=\begin{bmatrix}0 & -16\\-10 & -6\end{bmatrix}$

answered Apr 11, 2013
edited Dec 19, 2013