Find A, if 4 1 3 A= -4&8&4 -1&2&1 -3&6&3 . - Vidyadip

Question

Find A, if $\begin{bmatrix}4\\1\\3\end{bmatrix}\text{A}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}.$

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Answer

We have, $\begin{bmatrix}4\\1\\3\end{bmatrix}_{3\times1}\text{A}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}_{3\times3}$Let $\text{A}=\begin{bmatrix}\text{x}&\text{y}&\text{z}\end{bmatrix}$
$\therefore\ \begin{bmatrix}4\\1\\3\end{bmatrix}_{3\times1}\begin{bmatrix}\text{x}&\text{y}&\text{z}\end{bmatrix}_{1\times3}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}_{3\times3}$
$\Rightarrow\ \begin{bmatrix}4\text{x}&4\text{y}&4\text{z}\\\text{x}&\text{y}&\text{z}\\3\text{x}&3\text{y}&3\text{z}\end{bmatrix}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}$
$\Rightarrow\ 4\text{x}=-4\Rightarrow\ \text{x}=-1,\ 4\text{y}=8$
$\Rightarrow\ \text{y}=2\ \text{and }4\text{z}=4$
$\Rightarrow\ \text{z}=1$
$\therefore\ \text{A}=\begin{bmatrix}-1&2&1\end{bmatrix}$

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