1. The matrix A is:
[tex]A=\begin{bmatrix}{3} & {4} & {} \\ {1} & {-2} & {} \\ {} & {} & {}\end{bmatrix}[/tex]The determinant of a 2x2 matrix can be found as follows:
[tex]\begin{gathered} \det \begin{bmatrix}{a} & {b} & {} \\ {c} & {d} & {} \\ {} & {} & {}\end{bmatrix}=a\times d-b\times c \\ \text{Thus }\det |A|=3\times(-2)-4\times1 \\ \det |A|=-6-4 \\ \det |A|=-10 \end{gathered}[/tex]The determinant of matrix A is -10
2. The inverse of a matrix is given by:
[tex]\begin{gathered} A^{-1}=\frac{1}{\det |A|}\begin{bmatrix}{d} & {-b} & {} \\ {-c} & {a} & {} \\ {} & {} & {}\end{bmatrix} \\ A^{-1}=\frac{1}{-10}\begin{bmatrix}{-2} & {-4} & {} \\ {-1} & {3} & {} \\ {} & {} & {}\end{bmatrix} \\ A^{-1}=\begin{bmatrix}{\frac{-2}{-10}} & {\frac{-4}{-10}} & {} \\ {\frac{-1}{-10}} & {\frac{3}{-10}} & {} \\ {} & {} & {}\end{bmatrix}=\begin{bmatrix}{\frac{1}{5}} & {\frac{2}{5}} & {} \\ {\frac{1}{10}} & {-\frac{3}{10}} & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}[/tex]