If $P\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$, then matrix P is equal to:

$\begin{bmatrix}1&-2\\2&0\end{bmatrix}$

The correct answer is Option (2) → $\begin{bmatrix}1&-2\\2&0\end{bmatrix}$

$P\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$

Let $P=\begin{bmatrix}a&b\\c&d\end{bmatrix}$. Then

$\begin{bmatrix}a+4b&2a+5b&3a+6b\\c+4d&2c+5d&3c+6d\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$

$a+4b=-7,\ 2a+5b=-8\ \Rightarrow\ a+b=-1,\ b=-2,\ a=1$

$c+4d=2,\ 2c+5d=4\ \Rightarrow\ c+d=2,\ d=0,\ c=2$

$P=\begin{bmatrix}1&-2\\2&0\end{bmatrix}$