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If $X=\begin{bmatrix}3 & 1 & 1\\5 & 2 & 3\end{bmatrix}$ and $Y=\begin{bmatrix}2 & 1 & 1\\7 & 2 & 4\end{bmatrix}$, find a matrix $Z$ such that $X+Y+Z$ is a zero matrix.

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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 ≤ i ≤ m and 1 ≤ j ≤ n.
Given X+Y+Z=0
$\begin{bmatrix}3 & 1 & 1\\5 & 2 & 3\end{bmatrix}+\begin{bmatrix}2 & 1 & 1\\7 & 2 & 4\end{bmatrix}+Z=0$
Let us assume
$Z=\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}$
$\begin{bmatrix}3 & 1 & 1\\5 & 2 & 3\end{bmatrix}+\begin{bmatrix}2 & 1 & 1\\7 & 2 & 4\end{bmatrix}+\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}$=0
$\begin{bmatrix}3+2 & 1+1 & 1+1\\5+7 & 2+2 & 3+4\end{bmatrix}+\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}$=0
$\begin{bmatrix}5 & 2 & 2\\12 & 4 & 7\end{bmatrix}+\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}$=0
$\Rightarrow \begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}=(-1)\begin{bmatrix}5 & 2 & 2\\12 & 4 & 7\end{bmatrix}$
$\qquad\qquad\qquad\qquad=\begin{bmatrix}-5 & -2 & -2\\-12 &- 4 & -7\end{bmatrix}$
$Z=\begin{bmatrix}-5 & -2 & -2\\-12 &- 4 & -7\end{bmatrix}$
answered Mar 21, 2013

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