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If $\left[\begin{array}{cc}5 & 0 \\ 0 & 7\end{array}\right]^{-1}\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{c}-1 \\ 2\end{array}\right]$, then which one of the following is correct?
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Verified Answer
The correct answer is:
$x=-5, y=-14$
The given equation is : $\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]^{-1}\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{c}-1 \\ 2\end{array}\right]$
Multiplying both sides by $\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]$, we get
$\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]^{-1}\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]\left[\begin{array}{c}-1 \\ 2\end{array}\right]$
$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}-\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{l}-5 \\ 14\end{array}\right]$
$\Rightarrow\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{c}-5 \\ 14\end{array}\right] \quad \Rightarrow \mathrm{x}=-5$ and $\mathrm{y}=-14$
Multiplying both sides by $\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]$, we get
$\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]^{-1}\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 0 & 7\end{array}\right]\left[\begin{array}{c}-1 \\ 2\end{array}\right]$
$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}-\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{l}-5 \\ 14\end{array}\right]$
$\Rightarrow\left[\begin{array}{c}\mathrm{x} \\ -\mathrm{y}\end{array}\right]=\left[\begin{array}{c}-5 \\ 14\end{array}\right] \quad \Rightarrow \mathrm{x}=-5$ and $\mathrm{y}=-14$
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