Question
If $A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]$ and $B=\left[\begin{array}{l}7 \\ 0\end{array}\right]$, find matrix $C$ if $AC = B.$
✓
Answer
Let $C=\left[\begin{array}{l}a \\ b\end{array}\right]$
then $A C=B$
$\begin{array}{l}\Rightarrow\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}7 \\ 0\end{array}\right] \end{array} $
$ \Rightarrow\left[\begin{array}{c}3 a+b \\ -a+2 b\end{array}\right]=\left[\begin{array}{l}7 \\ 0\end{array}\right]$
$\Rightarrow 3a + b = 7 ...(1)$
$- a + 2b = 0 ...(2)$
From equation $(1),$
$6a + 2b = 14 ...(3)$
From $(3) - (2)$ given
$7a = 14$
$\Rightarrow a = 2$
Put $a = 2$ in $(1),$ we get
$6 + b = 7$
$\Rightarrow b = 7 - 6 = 1$
$\therefore C=\left[\begin{array}{l}2 \\ 1\end{array}\right]$.
then $A C=B$
$\begin{array}{l}\Rightarrow\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}7 \\ 0\end{array}\right] \end{array} $
$ \Rightarrow\left[\begin{array}{c}3 a+b \\ -a+2 b\end{array}\right]=\left[\begin{array}{l}7 \\ 0\end{array}\right]$
$\Rightarrow 3a + b = 7 ...(1)$
$- a + 2b = 0 ...(2)$
From equation $(1),$
$6a + 2b = 14 ...(3)$
From $(3) - (2)$ given
$7a = 14$
$\Rightarrow a = 2$
Put $a = 2$ in $(1),$ we get
$6 + b = 7$
$\Rightarrow b = 7 - 6 = 1$
$\therefore C=\left[\begin{array}{l}2 \\ 1\end{array}\right]$.
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