Answer
$A=\left[\begin{array}{cc}-2 & 1 \\ 4 & -1\end{array}\right]$
Work Step by Step
$A$ must be $2 \times 2$ matrix.
Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$
First vector:
$\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\left[\begin{array}{l}1 \\ 3\end{array}\right]=\left[\begin{array}{l}1 \\ 1\end{array}\right]$
$a+3 b=1$
$c+3 d=1$
Second vector:
$\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\left[\begin{array}{l}2 \\ 7\end{array}\right]=\left[\begin{array}{l}3 \\ 1\end{array}\right]$
$2 a+7 b=3$
$2 c+7 d=1$
First system:
$a+3 b=1$
$2 a+7 b=3$
Multiply first equation with -2 and add to second:
$2 a+7 b-2 a-6 b=3-2$
$b=1$
$a+3 b=1$
$a+3=1$
$a=-2$
Second system:
$c+3 d=1$
$2 c+7 d=1$
Multiply first equation with -2 and add to second:
$2 c+7 d-2 c-6 d=1-2$
$d=-1$
$c-3=1$
$c=4$
$A=\left[\begin{array}{cc}-2 & 1 \\ 4 & -1\end{array}\right]$
