Answer
$\begin{bmatrix} -\frac{-3}{2} & 1 \\ \frac{7}{2} & -2 \\ \end{bmatrix}$
Work Step by Step
The determinant for a $2 \times 2$ matrix $\begin{bmatrix} a & b \\c & d \\ \end{bmatrix}$ can be defined as:
$$det= \begin{bmatrix} a & b \\c & d \\ \end{bmatrix}=ad-bc$$
and the inverse of the matrix $A $ is given by: $A^{-1}=\dfrac{1}{|A|} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\quad \quad \text{(Equation 1)}$
We will evaluate the determinant of the given matrix.
$$|A|=\begin{bmatrix} 4 & 2 \\ 7 & 3 \\ \end{bmatrix}=(4)(3)-(2)(7)=-2$$
Plug $-2$ for $|A|$ and the respective values of $a,b,c$ and $d$ in $\text{Equation 1}$ above to obtain:
$$A^{-1}=\dfrac{1}{-2} \begin{bmatrix} 3 & -2 \\ -7 & 4 \\ \end{bmatrix}$$
Therefore, the inverse of $A$ is equal to $\begin{bmatrix} -\frac{-3}{2} & 1 \\ \frac{7}{2} & -2 \\ \end{bmatrix}$.
